Ruđer Bošković Institute · 4 Recurrence and transience of stable-like Markov chains 55 4.1...

SVEUCILISTE U ZAGREBUPRIRODOSLOVNO-MATEMATICKI FAKULTET

MATEMATICKI ODSJEK

Nikola Sandric

Recurrence and transience property of some Markov

chains

Disertacija

Voditelj rada: prof. dr. sc. Zoran Vondracek

Zagreb, 2012.

This thesis has been submitted for evaluation to the Department of Mathematics, Fac-

ulty of Science, University of Zagreb in fulfillment of the Degree of Doctor of Philosophy.

Ova disertacija je predana na ocjenu Matematickom odsjeku Prirodoslovno-matematickog

fakulteta Sveucilista u Zagrebu u svrhu stjecanja znanstvenog stupnja doktora prirodnih

znanosti iz podrucja matematike.

Contents

Contents i

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Related results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Preliminaries on stochastic processes 7

2.1 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Stable distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Markov models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Feller processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Characteristics of semimartingales . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Convergence of semimartingales . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Recurrence and transience of Markov models . . . . . . . . . . . . . . . . . 32

3 Stable-like Markov chains 43

3.1 General stable-like Markov chains . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Periodic stable-like Markov chains . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 (α, β)-stable-like Markov chains . . . . . . . . . . . . . . . . . . . . . . . . 51

i

4 Recurrence and transience of stable-like Markov chains 55

4.1 Recurrence of general stable-like Markov chains . . . . . . . . . . . . . . . 55

4.2 Transience of general stable-like Markov chains . . . . . . . . . . . . . . . . 66

4.3 Recurrence and transience of periodic stable-like Markov chains . . . . . . 72

4.4 Recurrence and transience of (α, β)-stable-like Markov chains . . . . . . . . 83

4.5 Applications and generalizations . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Discrete state case 95

5.1 Recurrence and transience of general discrete stable-like Markov chains . . 96

5.2 Recurrence and transience of discrete periodic stable-like Markov chains . . 98

5.3 Recurrence and transience of (α, β)-stable-like Markov chains . . . . . . . . 103

Bibliography 107

ii

Chapter 1

Introduction

A Markov model is a collection of random variables X = Xtt∈T, where T is a linearlyordered set (the time set), without memory, i.e., the future of the process is independentof the past given only its present value. According to this property, the usage of Markovmodels is wide, both in theory and applications. Markov models with discrete time andvalues in a discrete set (discrete Markov models) can nowadays be regarded as a part ofclassical probability theory. The analysis of general Markov models requires more sophis-ticated techniques than in the discrete case. The central problem considered in this thesisis the recurrence and transience property of Markov models. This is a property of stabilityand instability of Markov models. Transient (unstable) Markov models have a tendencyto return only finite number of times to a set of “reasonable size”, in other words, theyhave a tendency to move away from the center of the state space. On the other hand, re-current (stable) Markov models do the opposite, they have a tendency to visit any “nice”set infinitely many times, i.e., they have a tendency to turn back to the center of the statespace.

1.1 Motivation

Let (Ω,F ,P) be a probability space and let Ynn∈N be a sequence of i.i.d. randomvariables on (Ω,F ,P) taking values in Rd. Let us define Sn :=

∑ni=1 Yi and S0 := 0. The

sequence S = Snn∈Z+ is called a random walk. The random walk S is said to be recurrentif

P(

lim infn−→∞

|Sn| = 0)

= 1,

and transient ifP(

limn−→∞

|Sn| =∞)

= 1.

It is well known that every random walk is either recurrent or transient (see [12, Theorem4.2.1]). It is also well known that every truly d-dimensional random walk is transient ifd ≥ 3 (see [12, Theorem 4.2.13]). On the other hand, in the class of truly two-dimensionalstable random walks in R2, by [12, Theorem 4.2.9], the only recurrent case is the case whenS is a random walk with zero mean Gaussian jumps. In the case d = 1, a symmetric stablerandom walk, i.e., a random walk with jump distribution with characteristic exponent

1

2 1. Introduction

ψ(ξ) = γ|ξ|α, where α ∈ (0, 2] and γ ∈ (0,∞), is recurrent if, and only if, α ≥ 1 (see thediscussion after [12, Lemma 4.2.12]).

In this thesis, we generalize a one-dimensional symmetric stable random walk in the waythat the index of stability of jump distribution depends on the current position and westudy recurrence and transience of this generalized random walk. In other words, we areinterested in the recurrence and transience of a Markov chain with transition jumps withcharacteristic exponents ψ(x; ξ) = γ(x)|ξ|α(x), where α : R −→ (0, 2) and γ : R −→ (0,∞)are arbitrary functions.

1.2 Related results

A similar problem was firstly considered by Kemperman in [23]. More precisely, hestudied the recurrence and transience of a Markov chain on Z, the so-called oscillatingrandom walk, given by the following transition function

p(i, j) :=

µ(j − i), i < 0

pµ(j) + qν(j), i = 0ν(j − i), i > 0,

where µ(·) and ν(·) are arbitrary distributions on Z and p, q ∈ R+ are such that p +q = 1. Under certain mild conditions on the distributions µ(·) and ν(·) which guaranteeirreducibility of the chain, he derived necessary and sufficient condition for the recurrenceand transience in terms of the distributions µ(·) and ν(·). The oscillating random walk isrecurrent if, and only if,

∞∑h=1

C+µ (h)C−ν (h) =∞, (1.1)

where the numbers C+µ (h) and C−ν (h) are given by

C+µ (h) =

∞∑n=1

P(Sµn = h, Sm > 0 for 1 ≤ m ≤ n)

and

C−ν (h) =∞∑n=1

P(Sνn = −h, Sm < 0 for 1 ≤ m ≤ n),

and the processes Sµ and Sν denote the random walks with µ(·) and ν(·) as jump distri-butions, respectively.

Kemperman’s work was continued by Rogozin and Foss in [36]. Using (1.1), they deriveda necessary and sufficient condition, of the Chung-Fuchs type, for the recurrence andtransience of oscillating random walks. In the special case, when the random walks Sµ

and −Sν are strongly attracted to symmetric stable distributions with indices of stabilityα, β ∈ (0, 2), respectively, they proved that if α+ β > 2, then the oscillating random walkis recurrent, and if α + β < 2, then the oscillating random walk is transient (see Section5.3 for details).

1.3 Overview 3

A continuous-time analogue of the previous situation was considered by Bottcher in [7].He proved that the Feller process determined by a symbol of the form p(x, ξ) = γ(x)|ξ|α(x),called a stable-like process, is recurrent if, and only if, α + β ≥ 2, where the functionsα : R −→ (0, 2) and γ : R −→ (0,∞) are of class C1

b (R) and such that

α(x) =

α, x < −kβ, x > k

and γ(x) =

γ, x < −kδ, x > k

for some γ, δ, k > 0.In the case when the functions α : R −→ (0, 2) and γ : R −→ (0,∞) are continuously

differentiable and periodic, Franke in [15, 16] proved that if the set x ∈ R : α(x) = α0 :=infx∈R α(x) has positive Lebesgue measure, then the corresponding stable-like process isrecurrent if, and only if, α0 ≥ 1.

Schilling and Wang in [43] proved the transience of a stable-like process whose stabilityfunction α(x) is uniformly bounded above by 1. More precisely, they showed that if α :R −→ (0, 2) and γ : R −→ (0,∞) are of class C1

b (R) and if lim sup|x|−→∞ α(x) < 1, thenthe corresponding stable-like process is transient.

In this thesis, we consider only the case when α(x) ∈ (0, 2), i.e., a generalization of asymmetric α-stable random walk. Menshikov et al. in [27] treated the case of a Markovchain on R with uniformly bounded 2 + δ0-moments of transition jumps, for some δ0 > 0.Hence, if we alow that α(x) ∈ (0,∞), [27] covers the case when lim inf |x|−→∞ α(x) > 2.

1.3 Overview

In Chapter 2, we give some preliminary definitions, results and notation concerningstochastic processes. We refer to the text books [4], [9], [10], [12], [14], [18], [37], [45] and [47]for probability theory, [29], [32] and [34] for Markov chains theory, [5], [13], [19–21], [25], [35]and [40] for Markov processes theory and [22] and [33] for theory of semimartingales.

In Chapter 3, we introduce our main object of investigation, a stable-like Markov chain,defined by the following transition kernel

p(x, dy) := fx(y − x)dy,

where fxx∈R is a family of density functions on R satisfying

(i) x 7−→ fx(y) is a Borel measurable function for all y ∈ R;

(ii) fx(y) ∼ c(x)|y|−α(x)−1, when |y| −→ ∞,

for some functions α : R −→ (0, 2) and c : R −→ (0,∞). We also treat two specialcases of stable-like Markov chains: (i) periodic case, i.e., the case when x 7−→ fx is aperiodic function, and (ii) (α, β)-case, i.e., the case when fx(y) is the density function of asymmetric α-stable distribution for x < 0, and the density function of a symmetric β-stabledistribution for x ≥ 0, for α, β ∈ (0, 2). Furthermore, we prove Lebesgue irreducibility,aperiodicity, a sort of continuity and we determine a class of “nice” sets (petite sets) forthis class of Markov chains.

4 1. Introduction

Chapter 4 is devoted to the central problem of this thesis, the recurrence and transienceproperty of stable-like Markov chains. In Section 4.1, by using the Foster-Lyapunov driftcondition for recurrence, we prove that, under a uniformity condition on the density func-tions fxx∈R and some mild technical conditions, a stable-like Markov chain is recurrentif lim sup|x|−→∞ α(x) > 1. Similarly, in Section 4.2, we prove that a stable-like Markovchain is transient if lim inf |x|−→∞ α(x) < 1. Further, in Section 4.3 we prove that if theset x ∈ R : α(x) = α0 := infx∈R α(x) has positive Lebesgue measure, then the periodicstable-like Markov chain is recurrent if, and only if, α0 ≥ 1. In Section 4.4, we we provethat the (α, β) stable-like Markov chain is recurrent if, and only if, α + β ≥ 2.

Finally, in Section 5 we consider discrete state space version of stable-like Markov chainsand we derive the same recurrence and transience conditions as in the continuous statespace case.

1.4 Notation

By N, Z, R and C we denote, respectively, the sets of positive integers, integers, realnumbers and complex numbers. We write Z+, Z− and R+, respectively, for nonnegativeintegers, nonpositive integers and nonnegative real numbers. Furthermore, a ∧ b and a ∨ bdenote the minimum and maximum of a, b ∈ R, respectively. The expression sgn(x ), x ∈ R,represents the sign function, i.e.,

sgn(x ) =

−1, x < 00, x = 01, x > 0.

The indicator function of an arbitrary set A is defined by

1A(x) =

1, x ∈ A0, x /∈ A.

For d ∈ N, Rd denotes the d-dimensional Euclidean space. The components of x ∈ Rd

are denoted by xi ∈ R, i = 1, . . . , d. The inner product and norm on Rd are defined by〈x, y〉 :=

∑di=1 x

iyi and |x| := 〈x, x〉 12 , respectively. The components of an n×m, n,m ∈ N,real matrix M are denoted by mij ∈ R, for 1 ≤ i ≤ n and 1 ≤ j ≤ m. For a ∈ R, b ∈ Rd

and A ⊆ Rd we define aA := ax : x ∈ A and A+ b := x+ b : x ∈ A.For a topological space (T, T ) and arbitrary set A ⊆ T , the interior, closure and bound-

ary of A are denoted by Int(A), Cl(A) and ∂A, respectively. Furthermore, by B(T ) wedenote the Borel σ-algebra on (T, T ). For A ∈ B(T ), B(A) denotes the Borel σ-algebraof all Borel sets included in A. By λ(·) is denoted the Lebesgue measure on (Rd,B(Rd)).A probability measure on (Rd,B(Rd)) is usually called a distribution. The characteristicfunction of a distribution µ(·) is defined as follows

µ(ξ) :=

∫Rdei〈ξ,x〉µ(dx), ξ ∈ Rd.

A mapping between two measurable spaces f : (M,M) −→ (N,N ) is measurable

1.4 Notation 5

provided f−1(A) ∈ M for all sets A ∈ N . When there is no confusion about the domainand codomain of a measurable mapping f : (M,M) −→ (N,N ) we just say that f(x) isM/N measurable. For a family of mappings fi : M −→ (Ni,Ni), i ∈ I, the σ-algebragenerated by this family, i.e., the smallest σ-algebra on M such that all the mappingsfi, i ∈ I, are measurable, is denoted by σfi : i ∈ I. Let µ(·) and ν(·) be two positivemeasures on a measurable space (M,M). We say that ν(·) is absolutely continuous withrespect to µ(·), and write ν µ, if ν(B) = 0 for every B ∈ M such that µ(B) = 0. Let(M,M, µ) be a measure space, where µ(·) is a positive measure, and let p ∈ [1,∞). Foran M/B(R) measurable function f(x) define

||f ||p :=

(∫M

|f |pdµ) 1

p

,

[f ] := g : M −→ R : g(x) is M/B(R) measurable and g(x) = f(x) µ-a.e.

andLp(M,M, µ) := [f ] : f(x) is M/B(R) measurable and ||f ||p <∞.

Then, || · ||p is a norm on Lp(M,M, µ) and (Lp(M,M, µ), || · ||p) is a Banach space. Forf ∈ L1(Rd,B(Rd), λ) the Fourier transform of f(x) is defined as follows

f(ξ) := (2π)−d∫Rde−i〈ξ,x〉f(x)dx, ξ ∈ Rd.

By Bb(Rd) is denoted the space of bounded B(Rd)/B(R) measurable functions. Togetherwith the supremum norm ||·||∞ := supx∈Rd |·|, Bb(Rd) becomes a Banach space. By C(Rd),Cb(Rd), C0(Rd) and Cc(R) are denoted, respectively, the spaces of continuous functions,continuous bounded functions, continuous functions vanishing at infinity and continuousfunctions with compact support. The spaces Cb(Rd) and C0(Rd), together with the supre-mum norm || · ||∞, are Banach spaces. For k ∈ N ∪ ∞, Ck(Rd) denotes the space of ktimes differentiable functions, and Ck

b (Rd) and Ckc (Rd) denote the spaces of k times differ-

entiable functions such that all derivatives up to order k are bounded, respectively, withcompact support. A d-multi-index, d ∈ N, is an ordered d-tuple of nonnegative integers.For a d-multi-index α = (α1, . . . , αd) we set

∂α :=∂α1

∂(x1)α1· · · ∂αd

∂(xd)αd.

For f ∈ C∞(Rd), N ∈ Z+ and a d-multi-index α we define

||f ||(N,α) := supx∈Rd

(1 + |x|)N |∂αf(x)|.

The Schwartz space is the space

S(Rd) := f ∈ C∞(Rd) : ||f ||(N,α) <∞ for all N ∈ Z+ and all d-multi-indices α.

A function f : R −→ Rd is called cadlag (respectively, cag) if it is right continuous and has

6 1. Introduction

left limits (respectively, left continuous). The space of all cadlag functions f : R+ −→ Rd

is denoted by D(Rd).For two functions f, g : R −→ R we write f(x) = o(g(x)), f(x) = O(g(x)) and f(x) ∼

g(x), when x −→ x0, if limx−→x0 f(x)/g(x) = 0, lim supx−→x0 |f(x)|/|g(x)| < ∞ andlimx−→x0 f(x)/g(x) = 1, respectively, where x0 ∈ [−∞,∞].

Chapter 2

Preliminaries on stochastic processes

In this chapter, we recall some standard and well-known facts, results and notationregarding stochastic processes which will be used in the following chapters.

2.1 Special functions

The Gamma function is a function defined by Γ(z) :=∫∞

0e−ttz−1dt for z ∈ C, Re(z ) >

0. It satisfiesΓ(z + 1) = zΓ(z), (2.1)

and it can be analytically continued on z ∈ C \ Z−.The Digamma function is a function defined by Ψ(z) := Γ′(z)

Γ(z)for z ∈ C, Re(z ) > 0.

Lemma 2.1.1 Let a > 0 be an arbitrary real number. Then∫ ∞1

dy

ya(1 + y)=

1

2

(Ψ

(a+ 1

2

)−Ψ

(a2

)).

Proof. From [1, formula 6.3.22] we have

Ψ(z) =

∫ 1

0

1− xz−1

1− xdx− γ

for Re(z ) > 0, where γ is Euler’s constant. Thus,

Ψ

(a+ 1

2

)−Ψ

(a2

)=

∫ 1

0

xa2−1 − xa+1

2−1

1− xdx.

The claim now follows by change of variables x = y−2. 2

The Gauss hypergeometric function is defined by the formula

2F1(a, b, c; z) :=∞∑n=0

(a)n(b)n(c)n

zn

n!(2.2)

7

8 2. Preliminaries on stochastic processes

for a, b, c, z ∈ C, c /∈ Z−, where for w ∈ C and n ∈ Z+, (w)n is defined by

(w)0 = 1 and (w)n = w(w + 1) · · · (w + n− 1).

The series in (2.2) absolutely converges for |z| < 1, absolutely converges for |z| ≤ 1when Re(c − a − b) > 0, conditionally converges for |z| ≤ 1, except for z = 1, when−1 < Re(c − b − a) ≤ 0 and diverges when Re(c − b − a) ≤ −1. In the case whenRe(c) > Re(b) > 0, it can be analytically continued on C \ (1,∞) by the formula

2F1(a, b, c; z) =Γ(c)

Γ(b)Γ(c− b)

∫ 1

0

tb−1(1− t)c−b−1(1− tz)−adt. (2.3)

Let us list some properties of the Gauss hypergeometric function which will be needed inthe sequel (see [1, formulas 15.1.20, 15.3.3 and 15.3.7]):

(i) for a, b, c, z ∈ C, c /∈ Z−,

2F1(0, b, c; z) = 2F1(a, 0, c; z) = 1 (2.4)

(ii) for a, b, c, z ∈ C, Re(c − a − b) > 0, c /∈ Z−,

2F1(a, b, c; 1) =Γ(c)Γ(c− a− b)Γ(c− a)Γ(c− b)

(2.5)

(iii) for a, b, c, z ∈ C, z ∈ C \ (1,∞)

2F1(a, b, c; z) = (1− z)c−b−a 2F1(c− a, c− b, c; z) (2.6)

(iv) for a, b, c, z ∈ C, z ∈ C \ (0,∞)

2F1(a, b, c; z) =Γ(c)Γ(b− a)

Γ(b)Γ(c− a)(−z)−a 2F1

(a, 1− c+ a, 1− b+ a,

1

z

)+

Γ(c)Γ(a− b)Γ(a)Γ(c− b)

(−z)−b 2F1

(b, 1− c+ b, 1− a+ b,

1

z

). (2.7)

The incomplete Beta function is defined by the formula

B(x; z, w) :=

∫ x

0

tz−1(1− t)w−1dt (2.8)

for x ∈ [0, 1], z, w ∈ C, Re(z ) > 0 and Re(w) > 0. When x = 1, the function B(1; z, w) iscalled the Beta function and it satisfies

B(1; z, w) =Γ(z)Γ(w)

Γ(z + w). (2.9)

For further properties of the Gamma function, Digamma function, hypergeometric func-tions, incomplete Beta functions and Beta function see [1, Chapters 6 and 15].

2.2 Stable distributions 9

2.2 Stable distributions

A distribution µ(·) on B(Rd) is infinitely divisible if, for any n ∈ N, there is adistribution µn(·), such that

µ(ξ) = µnn(ξ) holds for all ξ ∈ Rd.

The celebrated Levy-Khintchine theorem gives a representation of characteristic functionsof all infinitely divisible distributions.

Theorem 2.2.1 [40, Theorem 8.1] A distribution µ(·) on B(Rd) is infinitely divisible if,and only if, there exist b ∈ Rd, symmetric nonnegative-definite d×d matrix c and measureν(·) on B(Rd) satisfying

ν(0) = 0 and

∫Rd

(1 ∧ |y|2)ν(dy) <∞,

such thatµ(ξ) = e−ψ(ξ) for ξ ∈ Rd,

where

ψ(ξ) = −i〈ξ, b〉+1

2〈ξ, cξ〉 −

∫Rd

(ei〈ξ,y〉 − 1− i〈ξ, y〉1z:|z|≤1(y)

)ν(dy). (2.10)

Furthermore, the above representation of µ(ξ) by b, c and ν(·) is unique. The measureν(·), triplet (b, c, ν) and function ψ(ξ) are called the Levy measure, Levy triplet andcharacteristic exponent of the distribution µ(·), respectively. 2

The representation in (2.10) is called the Levy-Khintchine representation. Let usremark that there is nothing special with the function 1y:|y|≤1(x) in the Levy-Khintchinerepresentation. It can be replaced by any h ∈ Bb(Rd) satisfying

(i) h(x) = 1 + o(|x|) as |x| −→ 0

(ii) h(x) = O(|x|−1) as |x| −→ ∞.

A special subclass of infinitely divisible distributions is a class of stable distributions.A distribution µ(·) on B(Rd) is said to be stable if, for any n ∈ N, there are an > 0 andbn ∈ Rd, such that

µn(ξ) = ei〈ξ,bn〉µ(anξ) holds for all ξ ∈ Rd.

Theorem 2.2.2 [40, Theorem 14.3] A distribution µ(·) on B(Rd) is a stable distributionif, and only if, there exists α ∈ (0, 2], called the index of stability of the distributionµ(·), such that

(i) if α = 2, then (b, c, 0) is the Levy triplet of µ(·), i.e., µ(·) is the normal distribution


(ii) if α ∈ (0, 2), then (b, 0, ν) is the Levy triplet of µ(·) with

ν(B) =

∫Sd−1

η(dζ)

∫ ∞0

1B(rζ)dr

rα+1

for all B ∈ B(Rd), where Sd−1 := x ∈ Rd : |x| = 1 and η(·) is a finite measure onB(Sd−1). 2

In the sequel, we will only consider one-dimensional stable distributions. In this case, theLevy-Khintchine representation can be rewritten in the following form.

Theorem 2.2.3 [47, Theorem C3] Let µ(·) be a stable distribution on B(R). Then

µ(ξ) =

exp[iδξ − γ|ξ|α exp(−1

2iπK(α)β t

|ξ|)], α 6= 1

exp[iδξ − γ|ξ|(1− iβ 2πξ|ξ| log |ξ|)], α = 1,

(2.11)

for α ∈ (0, 2], β ∈ [−1, 1], γ > 0, δ ∈ R and K(α) = 1− |1− α|. 2

The parameters β, γ and δ are called the skewness parameter, scale parameter andshift parameter, respectively. From now on, a one-dimensional stable distribution withparameters α, β, γ and δ is denoted by Sα(β, γ, δ). If β or δ are equal to zero, we usually donot write them in the previous notation. Note that if µ(·) is a one-dimensional symmetricstable distribution with α ∈ (0, 2), then µ(ξ) = e−γ|ξ|

αand the Levy triplet is given by(

0, 0, γα2α−1Γ(α+1

2)

π12 Γ(1− α

2)

dx

|x|α+1

). (2.12)

A one-dimensional symmetric stable distribution is abbreviated by SαS.Let µ(·) be a Sα(β, γ, δ) distribution. Since

∫R |µ(ξ)|dξ < ∞, by [12, Theorem 3.3.5],

µ(·) has a density function

f(y;α, β, γ, δ) =1

2π

∫Re−iξyµ(ξ)dξ.

As before, if β or δ are equal to zero, we usually do not write them in the previous notation.In the following theorem, we list several properties of stable densities we need.

Theorem 2.2.4 [47, Theorems 2.4.2, 2.5.1 and 2.5.4] [17, Theorem 1] Let f(y;α, β, γ, δ)be the density function of a Sα(β, γ, δ) distribution with parameters α ∈ (0, 2), β ∈ (−1, 1),γ > 0 and δ ∈ R. Then, we have

(i) f(y;α, β, γ, δ) =

γ−

1αf((y − δ)γ− 1

α ;α, β, 1, 0), α 6= 1γ−1f((y − δ)γ−1 − 2βπ−1 log γ; 1, β, 1, 0), α = 1

(ii) f(−y;α, β, 1, 0) = f(y;α,−β, 1, 0)

2.2 Stable distributions 11

(iii) for α < 1 and y > 0

f(y;α, β, 1, 0) =1

π

∞∑n=1

(−1)n+1 Γ(nα + 1)

n!sin(nπα

2(β + 1)

)y−nα−1

(iv) for α > 1, y −→∞ and N ∈ N

f(y;α, β, 1, 0) =1

π

N∑n=1

(−1)n+1 Γ(nα + 1)

n!sin(nπ

2(α + (2− α)β)

)y−nα−1

+O(y−(N+1)α−1)

(v) for α = 1, y −→∞ and N ∈ N

f(y; 1, β, 1, 0) =1

π

N∑n=1

1

n!Pn(log x)y−n−1 +O(y−N−2(log y)N),

where Pn(y) =∑n

l=0 rlnyl, and

rln =n∑

m=l

(−1)m−l(

nm

)(ml

)Γm−l(n+ 1)βm

(π(1 + β)

2

)n−msin

(π(n−m)

2

)

(vi) f(y;α, β, γ, δ) is bell-shaped, i.e., it is infinitely time differentiable on R and itsk-th derivative f (k)(y;α, β, γ, δ) possesses exactly k zeros and they are simple. 2

A distribution µ(·) on B(R) belongs to the domain of attraction of a distribution functionπ(·), or it is attracted to a distribution function π(·), if there exist sequences of realnumbers ann∈N and bnn∈N, an > 0 for all n ∈ N, such that

limn−→∞

e−ibnξµn(ξ/an) = π(ξ) holds for all ξ ∈ R.

Furthermore, if bn = 0 for all n ∈ N, then we say that the distribution µ(·) is stronglyattracted to the distribution π(·). It is well known that the distribution π(·) can only bea Sα(β, γ, δ) distribution (see [9, Proposition 9.25]).

Theorem 2.2.5 [18, Theorem 35.2] A distribution µ(·) on B(R) belongs to the domain ofattraction of a Sα(β, γ, δ) distribution, with 0 < α < 2, if and only if there exist constantsc1, c2 ∈ R+, c1 + c2 > 0, such that

µ((−∞,−x])

µ((x,∞))−→ c1

c2

andµ((−∞,−x] ∪ (x,∞))

µ((−∞,−kx] ∪ (kx,∞))−→ kα

for all k > 0, as x −→∞. 2


In the proof of Theorem 2.2.5, the centering sequence bnn∈N is given explicitly by

bn =

0, 0 < α < 1

Im (log µ(1/an)) , α = 1nan

∫R yµ(dy), 1 < α < 2.

(2.13)

Note that∫R |y|µ(dy) <∞ in the case when 1 < α < 2 (see [9, Proposition 9.39]). Hence,

in order to get the strong attraction it is sufficient that µ(·) is a symmetric distribution, ifα = 1, and

∫R yµ(dy) = 0, if 1 < α < 2.

2.3 Markov models

Let (Ω,F ,P) be a probability space and let T be the time set R+ or Z+. A stochasticprocess with values in Rd and parameter set T is a family Xtt∈T of F/B(Rd) measurablerandom variables.

A stochastic process Ytt∈T is called a modification of a stochastic process Xtt∈T ifP(Xt = Yt) = 1 for all t ∈ T. Note that if Xtt∈T and Ytt∈T are modifications of eachother, then they have the same finite-dimensional distributions.

A stochastic process Xtt∈R+ is called a cadlag process (respectively, cag process) ifthere exists Γ ∈ F with P(Γ) = 0, such that t 7−→ Xt(ω) is a cadlag function (respectively,cag function) for all ω ∈ Γc. By setting

Xt(ω) =

Xt(ω), ω ∈ Γc

0 ω ∈ Γ,

the stochastic process Xtt∈R+ is a cadlag process for all ω ∈ Ω and P(Xt = Xt) = 1 forall t ∈ R+.

A family F = Ftt∈T of increasing sub-σ-algebras of F , i.e., Fs ⊆ Ft ⊆ F for s ≤ t,s, t ∈ T, is called a filtration of (Ω,F) and (Ω,F ,F) is called a filtered space. Astochastic basis is a probability space (Ω,F ,P) equipped with a filtration F.

Let (Ω,F ,P) be a probability space. A completion FP of F , with respect to P(·), is theσ-algebra of subsets B ⊆ Ω, such that there exist sets B1, B2 ∈ F with B1 ⊆ B ⊆ B2 andP(B2 \ B1) = 0. A stochastic basis (Ω,F ,P,F) satisfies the usual conditions if F = FP,F0 contains all P-null sets of F and the filtration F is right continuous, i.e., Ft = Ft+ forall t ∈ T, where

Ft+ :=⋂s>t

Fs.

A stochastic process Xtt∈T is adapted to a filtration F if the random variables Xt areFt/B(Rd) measurable for all t ∈ T. The natural filtration of a stochastic process Xtt∈Tis the filtration F0 = F0

t := σXs : s ∈ T, s ≤ t. Clearly, every stochastic process isadapted to its corresponding natural filtration.

Definition 2.3.1 Let (Ω,F ,P,F) be a stochastic basis and let Xtt∈T be a stochasticprocess adapted to the filtration F. The structure X = (Ω,F ,P,F, Xtt∈T) is called an

2.3 Markov models 13

elementary Markov model if the elementary Markov property holds, i.e.,

E[f(Xt)|Fs] = E[f(Xt)|Xs] P-a.s. (2.14)

holds for all s, t ∈ T, 0 ≤ s ≤ t and all f ∈ Bb(Rd).

Let us remark that here and in the sequel we use the term Markov model when the resultholds regardless of the time set involved. If T = R+ we replace the term Markov model bythe term Markov process, and if T = Z+ we replace it by the term Markov chain.

Definition 2.3.2 A family p = ps,t(x,B) : s, t ∈ T, 0 ≤ s ≤ t, defined for x ∈ Rd andB ∈ B(Rd), is called a Markov transition function if

(i) B 7−→ ps,t(x,B) is a probability measure for all x ∈ Rd and all s, t ∈ T, 0 ≤ s ≤ t

(ii) pt,t(x, ·) = δx(·) for all x ∈ Rd and all t ∈ T

(iii) x 7−→ ps,t(x,B) is B(Rd)/B(R) measurable for all B ∈ B(Rd) and all s, t ∈ T,0 ≤ s ≤ t

(iv) the Chapman-Kolmogorov equation stands, i.e., for all x ∈ Rd, B ∈ B(Rd) andall s, t, u ∈ T, 0 ≤ s ≤ t ≤ u,

ps,u(x,B) =

∫Rdps,t(x, dy)pt,u(y,B). (2.15)

Definition 2.3.3 Let (Ω,F ,P,F) be a stochastic basis and let Xtt∈T be a stochasticprocess adapted to the filtration F. Furthermore, let p be a Markov transition function.The structure X = (Ω,F ,P,F, Xtt∈T,p) is called a Markov model if the Markovproperty holds, i.e.,

E[f(Xt)|Fs] =

∫Rdf(y)ps,t(Xs, dy) P-a.s. (2.16)

holds for all s, t ∈ T, 0 ≤ s ≤ t, and all f ∈ Bb(Rd).

Taking conditional expectation with respect to σ(Xs) in (2.16) we get

E[f(Xt)|Xs] =

∫Rdf(y)ps,t(Xs, dy) P-a.s.,

i.e., combining with (2.16),

E[f(Xt)|Fs] = E[f(Xt)|Xs] P-a.s.

Therefore, every Markov model is an elementary Markov model.Let X be a Markov model and let µ(·) be the distribution of the random variable X0,

i.e., µ(·) = P(X0 ∈ ·). The distribution µ(·) is called the initial distribution of X. From


the Markov property we get the exact formula for the finite-dimensional distributions ofX. Indeed,

P(X0 ∈ B0, Xt1 ∈ B1, . . . , Xtn ∈ Bn)

=

∫B0

µ(dx0)

∫B1

p0,t1(x0, dx1)

∫B2

pt1,t2(x1, dx2) . . .

∫Bn

ptn−1,tn(xn−1, dxn)

for all n ∈ N and t1, . . . , tn ∈ T, 0 < t1 ≤ . . . ≤ tn, and all B0, B1, . . . , Bn ∈ B(Rd).In the sequel, we will be only interested in the case when a Markov transition function

p is temporally homogeneous, i.e., when ps,t(x,B) = pt−s(x,B) holds for some functionpt(x,B) and all x ∈ Rd, B ∈ B(Rd) and all s, t ∈ T, 0 ≤ s ≤ t.

Let (Ω,F ,F) be a filtered space. A mapping T : Ω −→ T ∪ ∞ is called a stoppingtime if T ≤ t ∈ Ft for all t ∈ T. For a stopping time T let us define FT := H ∈ F :H ∩ T ≤ t ∈ Ft for all t ∈ T and F ′T := H ∈ FT : H ⊆ T < ∞. It is easy to seethat FT and F ′T are σ-algebras.

Example 2.3.4 Let B ∈ B(Rd) and let the stochastic basis (Ω,F ,P,F) satisfy the usualconditions. If Xtt∈T is an F-adapted cadlag stochastic process, then the first hittingtime τB := inft ∈ T : Xt ∈ B is a stopping time (see [22, Theorems I.1.27 and I.1.54]).

Let T be a stopping time and let Xtt∈T be a stochastic process. DefineXT (ω) := XT (ω)(ω)on T <∞.

Definition 2.3.5 A Markov model X is a strong Markov model if for each stoppingtime T ,

(i) XT is F ′T/B(Rd) measurable

(ii) the strong Markov property holds, i.e.,

E[f(XT+t)|F ′T ] =

∫Rdf(y)pt(XT , dy) P(·|T <∞)-a.s. (2.17)

for all t ∈ T and all f ∈ Bb(Rd).

Note that if (Ω,F ,P,F, Xtt∈T,p) is a (strong) Markov model, then (Ω,F ,P,F0, Xtt∈T,p)is also a (strong) Markov model.

Remark 2.3.6 If a Markov process X has cadlag paths, then for any stopping time T therandom variable XT is F ′T/B(Rd) measurable.

Let X be a Markov model. We define a family of mappings θtt∈T on Ω, called shiftoperators, by Xs θt := Xs+t for all s, t ∈ T. If a family of shift operators exists, forthe Markov model X, then for all B ∈ B(Rd) and all s, t ∈ T we have θ−1

t X−1s (B) =

X−1s+t(B) ∈ Fs+t. Hence, θt is Fs+t/Fs measurable for all s, t ∈ T. In particular θt is

σXs : s ∈ T/σXs : s ∈ T measurable for all t ∈ T. Furthermore, using the shiftoperators, the Markov and strong Markov property can be written in the following form

E[f(Xt−s) θs|Fs] =

∫Rdf(y)pt−s(Xs, dy) P-a.s. (2.18)

2.3 Markov models 15

and

E[f(Xt) θT |F ′T ] =

∫Rdf(y)pt(XT , dy) P(·|T <∞)-a.s. (2.19)

for all s, t ∈ T, 0 ≤ s ≤ t.In the case T = Z+, by Chapman-Kolmogorov equation, Markov transition function p is

reduced just to one-step transition function p(x,B) := p1(x,B) for x ∈ Rd and B ∈ B(Rd).If µ(·) is the initial distribution of a Markov chain X, then

P(X0 ∈ B0, X1 ∈ B1, . . . , Xn ∈ Bn)

=

∫B0

µ(dx0)

∫B1

p(x0, dx1)

∫B2

p(x1, dx2) . . .

∫Bn

p(xn−1, dxn)

holds for all n ∈ N and all B0, B1, . . . , Bn ∈ B(Rd). It easy to see that when a stoppingtime takes only countably many values, the strong Markov property holds for any Markovmodel. Therefore, every Markov chain is actually a strong Markov chain.

Let (Ω,F ,P) be a probability space and let Ynn∈N be a sequence of i.i.d. randomvariables on (Ω,F ,P) taking values in Rd. Let us define Sn :=

∑ni=1 Yi and S0 := 0. The

sequence S = Snn∈Z+ is called a random walk. Clearly, any random walk S is a Markovchain with respect to the natural filtration and transition function p(x,B) := P(S1 ∈ B−x)for x ∈ Rd and B ∈ B(Rd).

Given a distribution µ(·) on B(Rd) and a transition function p, using Kolmogorov ex-tension theorem, we can construct a Markov model X with initial distribution µ(·) andtransition function p in the following way. Define Ω := (Rd)T and F0

∞ := (B(Rd))T. Foreach t ∈ T, let Xt : Ω −→ Rd be the coordinate mapping Xt(ω) := ω(t) and let F0

be the natural filtration. Clearly, F0∞ = σXt : t ∈ T. Let n ∈ N and t1, . . . , tn ∈ T,

0 < t1 ≤ . . . ≤ tn, be arbitrary. We define a probability measure on (B(Rd))n+1 by

Pµt1,...,tn(B0, B1, . . . , Bn)

=

∫B0

µ(dx0)

∫B1

pt1(x0, dx1)

∫B2

pt2−t1(x1, dx2) . . .

∫Bn

ptn−tn−1(xn−1, dxn),

where B0, B1, . . . , Bn ∈ B(Rd). By Kolmogorov’s extension theorem, there exists a uniqueprobability measure Pµ(·) on (Ω,F0

∞), such that

Pµ(X0 ∈ B0, Xt1 ∈ B1, . . . , Xtn ∈ Bn) = Pµt1,...,tn(B0, B1, . . . , Bn)

holds for all n ∈ N and t1, . . . , tn ∈ T, 0 < t1 ≤ . . . ≤ tn, and all B0, B1, . . . , Bn ∈ B(Rd).Using the Chapman-Kolmogorov equation, we get

Eµ[f(Xt)|Fs] =

∫Rdf(y)pt−s(Xs, dy) Pµ-a.s.

for all s, t ∈ T, 0 ≤ s ≤ t, and all f ∈ Bb(Rd). This Markov model is called the canonicalMarkov model. Note that any Markov model X = (Ω,F ,P,F, Xtt∈T,p) has the samefinite-dimensional distribution as the canonical Markov model constructed from the initialdistribution µ(·) := P(X0 ∈ ·) and the Markov transition function p.


If we fix x ∈ Rd and put µ(·) = δx(·), we get Px(Xt ∈ B) = pt(x,B). By [35, PropositionIII.1.6], the function x 7−→ Px(Γ) is B(Rd)/B(R) measurable for all Γ ∈ F and the Markovproperty reads

Ex[f(Xt)|Fs] = EXs [f(Xt−s)] Px-a.s.

for all s, t ∈ T, 0 ≤ s ≤ t, and all f ∈ Bb(Rd). Note that for any distribution µ(·) onB(Rd), for the canonical Markov model, we have

Pµ(·) =

∫Rd

Px(·)µ(dx)

and the Markov and strong Markov property can be rewritten as

Eµ[f(Xt)|Fs] = EXs [f(Xt−s)] Pµ-a.s.

andEµ[f(XT+t)|F ′T ] = EXT [f(Xt)] P(·|T <∞)-a.s.,

respectively. Also note that in the case of the canonical Markov model the shift operatorsalways exist and they are defined by θt(ω)(s) := ω(s+ t) for all ω ∈ Ω and all s, t ∈ T.

Motivated by the canonical Markov model we define a universal Markov model.

Definition 2.3.7 The structure X = (Ω,F , Pxx∈Rd ,F, Xtt∈T, θtt∈T) is called a uni-versal (strong) Markov model if the following conditions hold

(i) Xtt∈T is a stochastic process adapted to the filtration F

(ii) (Ω,F ,Px) is a probability space for all x ∈ Rd

(iii) the function x 7−→ Px(Γ) is B(Rd)/B(R) measurable for all Γ ∈ F

(iv) Px(X0 = x) = 1 for all x ∈ Rd

(v) Px(Xt ∈ Rd) = 1 for all t ∈ T

(vi) the mappings θt : Ω −→ Ω, called shift operators (with respect to the stochasticprocess Xt∈T), are F/F measurable and satisfy Xs θt = Xs+t for all s, t ∈ T

(vii) the universal Markov property holds, i.e.,

Ex[f(Xt−s θs)|Fs] = EXs [f(Xt−s)] Px-a.s. (2.20)

holds for all x ∈ Rd, all s, t ∈ T, 0 ≤ s ≤ t, and all f ∈ Bb(Rd)

(viii) (for the universal strong Markov model case) for each stopping time T , the randomvariable XT is F ′T/B(Rd) measurable and the universal strong Markov propertyholds, i.e.,

Ex[f(XT θt)|F ′T ] = EXT [f(Xt)] Px(·|T <∞)-a.s. (2.21)

holds for all x ∈ Rd, all t ∈ T and all f ∈ Bb(Rd).

2.4 Feller processes 17

Let us remark that for each x ∈ Rd the structure (Ω,F ,Px,F, Xtt∈T) is an elementaryMarkov model. Furthermore, it easy to see that the family p = pt(x,B) := Px(Xt ∈ B) :t ∈ T, defined for x ∈ Rd and B ∈ B(Rd), is a Markov transition function, and for eachx ∈ Rd the structure (Ω,F ,Px,F, Xtt∈T,p) is a (strong) Markov model.

Let µ(·) be an arbitrary distribution on B(Rd). Let us define the probability measureon (Ω,F) by

Pµ(·) :=

∫Rd

Px(·)µ(dx).

Clearly,

Eµ[f(Xt)|Fs] = EXs [f(Xt−s)] =

∫Rdf(y)pt−s(Xs, dy) Pµ-a.s. (2.22)

andEµ[f(XT θt)|F ′T ] = EXT [f(Xt)] Pµ(·|T <∞)-a.s. (2.23)

hold for all s, t ∈ T, 0 ≤ s ≤ t, and all f ∈ Bb(Rd). Therefore, for any distribution µ(·) onB(Rd), the structure (Ω,F ,Pµ,F, Xtt∈T,p) is a (strong) Markov model and the universal(strong) Markov property is equivalent with (2.22) ((2.23)).

Let X = (Ω,F , Pxx∈Rd ,F, Xtt∈T, θtt∈T) be a universal (strong) Markov model. Fora distribution µ(·) on B(Rd) we denote by Fµ the completion of F with respect to Pµ(·).Furthermore, let Fµt t∈T be the filtration obtained by adding to each Ft the Pµ-nullsetsN µ of Fµ, i.e., Fµt := σFt,N µ. Finally, we put

F c :=⋂µ

Fµ∞, F ct :=⋂µ

Fµt and Fc := F ct : t ∈ T,

where the intersection is taken over all distributions µ(·) on B(Rd). Then, by [21, Lemmas3.5.11 and 3.5.12], the structure Xc = (Ω,F c, Pxx∈Rd ,Fc, Xtt∈T, θtt∈T) is a universal(strong) Markov model.

2.4 Feller processes

Let (B, || · ||) be a Banach space and let T be the parameter (time) set R+ or Z+. Afamily of linear operators Ptt∈T on B is called a semigroup provided

(i) Pt : B −→ B is a bounded linear operator for all t ∈ T

(ii) P0 = I

(iii) Ps Pt = Ps+t for all s, t ∈ T, i.e., the family Pt∈T has the semigroup property.

Definition 2.4.1 Let Ptt∈T be a semigroup on a Banach space (B, || · ||). When T = R+,the infinitesimal generator of the semigroup Ptt∈R+ is a linear operator A : DA −→ Bdefined by

A(x) := limt−→0

Pt(x)− xt

,


where

DA :=

x ∈ B : lim

t−→0

Pt(x)− xt

exists

.

The set DA is called the domain of the infinitesimal generator A. When T = Z+, theinfinitesimal generator A : B −→ B of the semigroup Pnn∈Z+ is defined by A :=P1 − I.

Let X be a universal Markov model and, for each t ∈ T, let us define Ptf(x) :=Ex[f(Xt)] =

∫Rd f(y)pt(x, dy) for f ∈ Bb(Rd). It is easy to see that such defined family

Ptt∈T is a semigroup on (Bb(Rd), || · ||∞). Note that the semigroup Ptt∈T has threeadditional properties

(iv) ||Ptf ||∞ ≤ ||f ||∞ for all f ∈ Bb(Rd), i.e., Pt is a contraction for all t ∈ T

(v) it is conservative, i.e., Pt1 = 1 for all t ∈ T

(vi) it is a positivity preserving semigroup, i.e., Ptf ≥ 0 for all t ∈ T, wheneverf ≥ 0.

Definition 2.4.2 Let X be a universal Markov process and let Ptt∈R+ be its associatedsemigroup. If the semigroup Ptt∈R+ satisfies

(i) Pt(C0(Rd)) ⊆ C0(Rd) for all t ∈ R+, i.e., the C0-Feller property

(ii) limt−→0 Ptf = f for all f ∈ C0(Rd), i.e., the strong continuity property,

we call Ptt∈R+ a Feller semigroup and X is called a Feller process. If Pt(Cb(Rd)) ⊆Cb(Rd) for all t ∈ R+, then we say that the semigroup Ptt∈R+ satisfies the Cb-Fellerproperty, and if Pt(Bb(Rd)) ⊆ Cb(Rd) for all t > 0, we say that the semigroup Ptt∈R+

satisfies the strong Feller property.

Let us remark here that if we start from a Feller semigroup Ptt∈R+ then, by Riesz repre-sentation theorem, we can construct a Feller process X such that Ptf(x) = Ex[f(Xt)] forall t ∈ R+ and all f ∈ Bb(Rd) (see [19, Theorem 4.8.1]).

Theorem 2.4.3 [21, Theorem 3.4.19] Any Feller process has a cadlag modification. 2

Corollary 2.4.4 [21, Corollary 3.4.20] Any canonical Feller process has a cadlag modifi-cation. 2

Since we deal with universal Markov processes, Theorem 2.4.3 states that there existsa cadlag stochastic process Xtt∈R+ such that Pµ(Xt = Xt) = 1 for all t ∈ R+ and alldistributions µ(·) on B(Rd). Note that in this case, (Ω,F , Pxx∈Rd ,F, Xtt∈R+ , θtt∈R+),in general, is not a universal Markov process.

Let Xtt∈R+ be a cadlag modification of a Feller process X = (Ω,F , Pxx∈Rd ,F, Xtt∈R+ , θtt∈R+). For a distribution µ(·) on B(Rd) we denote by Fµ∞ the comple-tion of F0

∞ := σXt : t ∈ R+ with respect to Pµ(·), and by Fµ := Fµt t∈R+ the filtration


obtained by adding to each F0t := Xs : s ∈ R+, s ≤ t the Pµ-nullsets N µ of Fµ∞, i.e.,

Fµt := σF0t ,N µ. Finally we put

F c∞ :=⋂µ

Fµ∞, F ct :=⋂µ

Fµt and Fc := F ct t∈R+

where the intersection is taken over all distributions µ(·) on B(Rd).

Theorem 2.4.5 [21, Theorems 3.5.10, 3.5.13 and 3.5.14]

(i) The filtrations Fµ and Fc satisfy the usual conditions.

(ii) The process (Ω,F c∞, Pxx∈Rd ,Fc, Xtt∈R+ , θtt∈R+) is a Feller process and it satis-fies the universal strong Markov property. 2

Theorem 2.4.6 [21, Theorem 3.4.25] Every Feller process X is stochastically contin-uous, i.e.,

lims−→t

Pµ(|Xs −Xt| > ε) = 0

holds for all ε > 0, all t ∈ R+ and all distributions µ(·) on B(Rd). 2

A function u : Rd −→ C is called positive definite if for all n ∈ N, all ξ1, . . . , ξn ∈ Rd

and all λ1, . . . , λn ∈ C we have

n∑i,j=1

u(ξi − ξj)λiλj ≥ 0.

Definition 2.4.7 A function ψ : Rd −→ C is called negative definite if ψ(0) ≥ 0 andthe function ξ 7−→ e−tψ(ξ) is positive definite for all t ∈ R+.

Theorem 2.4.8 [19, Theorem 3.7.7] A function ψ : Rd −→ C is continuous negativedefinite if, and only if, it has the following Levy-Khintchine representation

ψ(ξ) = a− i〈ξ, b〉+1

2〈ξ, cξ〉 −

∫Rd

(ei〈ξ,y〉 − 1− i〈ξ, y〉1z:|z|≤1(y)

)ν(dy), (2.24)

where a ≥ 0, b ∈ Rd, c is a symmetric nonnegative-definite d × d matrix and ν(·) is ameasure on B(Rd), called the Levy measure, satisfying

ν(0) = 0 and

∫Rd

(1 ∧ |y|2)ν(dy) <∞.

The quadruple (a, b, c, ν) is called the Levy characteristics. 2

Note that a distribution µ(·) on B(Rd) is infinitely divisible if, and only if, the function ξ 7−→µt(ξ) is positive definite for all t ∈ R+. As in the case of infinitely divisible distributions,


we can replace the function 1y:|y|≤1(x) in the Levy-Khintchine representation by anyh ∈ Bb(Rd) satisfying

(i) h(x) = 1 + o(|x|) as |x| −→ 0

(ii) h(x) = O(|x|−1) as |x| −→ ∞.

An operator p(x,D) on the space S(Rd) is called a pseudo-differential operator if itcan be written as

p(x,D)f(x) = −∫Rdp(x, ξ)ei〈ξ,x〉f(ξ)dξ

for f ∈ S(Rd), where the function p : Rd × Rd −→ C has the following properties

(i) it is locally bounded in (x, ξ)

(ii) the function x 7−→ p(x, ξ) is B(Rd)/B(Rd) measurable for every ξ ∈ Rd

(iii) the function ξ 7−→ p(x, ξ) is a continuous negative definite function for every x ∈ Rd.

The function p(x, ξ) is called the symbol of the operator p(x,D). By Theorem 2.4.8, thesymbol has the representation

p(x, ξ) = a(x)− i〈ξ, b(x)〉+1

2〈ξ, c(x)ξ〉 −

∫Rd

(ei〈ξ,y〉 − 1− i〈ξ, y〉1z:|z|≤1(y)

)ν(x, dy),

(2.25)where a(x) is a nonnegative B(Rd)/B(R) measurable function, b(x) is a B(Rd)/B(Rd)measurable function, c(x) is a symmetric nonnegative-definite d × d matrix-valuedB(Rd)/B(Rd+d) measurable function and ν(x, ·) is a Borel nonnegative kernel on Rd×B(Rd)such that for every x ∈ Rd,

ν(x, 0) = 0 and

∫Rd

(1 ∧ |y|2)ν(x, dy) <∞.

Theorem 2.4.9 [11, Theorem 3.4] Let A : DA −→ C0(Rd) be the infinitesimal gener-ator of a Feller process. If C∞c (Rd) ⊆ DA, then A|C∞c (Rd) is a pseudo-differential operator.2

Theorem 2.4.10 [43, Theorem 1.1] Let X be an Rd valued Feller process with infinites-imal generator A : DA −→ C0(Rd), such that C∞c (Rd) ⊆ DA, and symbol p(x, ξ) given by(2.25). Let us assume that ||p(·, ξ)||∞ ≤ c(1 + |ξ|2) for all ξ ∈ Rd, for some c ≥ 0, andp(x, 0) = 0. Then the corresponding semigroup of X satisfies the Cb-Feller property andstrong Feller property. 2

Let us remark that the condition ||p(·, ξ)||∞ ≤ c(1 + |ξ|2) is equivalent with the boundnessof the Levy quadruple (a, s, c, ν), i.e., with

||a||∞ + ||b||∞ + ||c||∞ +

∣∣∣∣∣∣∣∣∫Rd

(1 ∧ |y|2)ν(·, dy)

∣∣∣∣∣∣∣∣∞<∞


(see [42, Lemma 2.1]), and the condition p(x, 0) = 0 (together with the assumption||p(·, ξ)||∞ ≤ c(1 + |ξ|2)) implies that the corresponding Feller process is conservative.If the function x 7−→ p(x, 0) is continuous, the condition p(x, 0) = 0 is also necessary forthe conservativeness of the corresponding Feller process (see [41, Theorem 5.2]).

Let µ(·) be an infinitely divisible distribution on B(Rd) with µ(ξ) = e−ψ(ξ), where

ψ(ξ) = −i〈ξ, b〉+1

2〈ξ, cξ〉 −

∫Rd

(ei〈ξ,y〉 − 1− i〈ξ, y〉1z:|z|≤1(y)

)ν(dy).

For each t ∈ R+ let µt(·) be an infinitely divisible distribution on B(Rd) with µt(ξ) = e−tψ(ξ)

(constructed as in [40, Lemma 7.9]). Furthermore, define the function p = pt(x,B) :=µt(B − x)t∈R+ for x ∈ Rd and B ∈ B(Rd). It is easy to see that p is a Markov transitionfunction. Therefore, it defines an Rd-valued canonical universal Markov process X. Definea family of linear operators Ptt∈R+ on Bb(R) by

Ptf(x) := Ex[f(Xt)] =

∫Rdf(y)pt(x, dy).

From infinitely divisibility property of the family µtt∈R+ , it follows easily that X isactually a (canonical) Feller process. Note also that above defined semigroup Ptt∈R+

satisfies the Cb-Feller property.

Definition 2.4.11 Let Xt∈T be a Rd valued stochastic process on the filtered probabilityspace (Ω,F ,P,F) adapted to the filtration F.

(i) We say that the process Xt∈T has independent increments if for all s, t ∈ T,0 ≤ s < t the random variable Xt −Xs is independent of Fs.

(ii) We say that the process Xt∈T has stationary increments if for all s, t ∈ T,0 ≤ s < t it follows that P(Xt −Xs ∈ ·) = P(Xt−s ∈ ·)

Theorem 2.4.12 [21, Theorem 3.7.2] Let µ(·) be an infinitely divisible distribution onB(Rd). Then, for any distribution η(·) on B(Rd), the above constructed (canonical) Fellerprocess has stationary and independent increments with respect to Pη(·). 2

Motivated with the above theorem we have the following.

Definition 2.4.13 A Feller process X is called a Levy process if

(i) Xtt∈R+ has independent increments

(ii) Xtt∈R+ has stationary increments with respect to Px(·) for all x ∈ Rd.

Let η(·) be an arbitrary distribution on B(Rd). For each t ∈ R+ define µt(·) :=Pη(Xt −X0 ∈ ·). Clearly, µt(·) is independent of the initial distribution η(·). Since X is aLevy process, µtt∈R+ is a family of infinitely divisible distributions on B(Rd) constructedfrom the distribution µ1(·) as before. Clearly, the canonical Levy process constructed fromthe family µtt∈R+ has the same finite dimensional distributions as the Levy process X.


Therefore, the distribution µ1(·), i.e., the corresponding Levy triplet, completely charac-terizes the Levy process X up to finite-dimensional distributions. Hence, µt(ξ) = e−tψ(ξ),where ψ(ξ) is the characteristic exponent of µ1(·) with the Levy triplet (b, c, ν) as above.Let us define a linear operator Ψ : S(Rd) −→ C0(Rd) by

Ψf(x) := −∫Rdψ(ξ)ei〈ξ,x〉f(ξ)dξ.

Next, for f ∈ S(Rd) we have

Ptf(x) =

∫Rdf(y)pt(x, dy) =

∫Rdf(y + x)µt(dy) =

∫Rde−tψ(ξ)ei〈ξ,x〉f(ξ)dξ.

Thus, the infinitesimal generator A of the Levy process X satisfies A|S(Rd) = Ψ, i.e.,the characteristic exponent ψ(ξ) and the quadruple (0, b, c, ν) are the symbol and Levyquadruple of the Levy process X, respectively.

Example 2.4.14 (i) Let d = 1 and let µ(·) be the Poisson distribution with parameterκ > 0. The corresponding Levy process is called a Poisson process. The symbol ofa Poisson process is given by p(ξ) = κ

(1− eiξ

)and the Levy triplet by (κ, 0, κδ1).

(ii) Let µ(·) be a stable distribution. The corresponding Levy process is called a stableLevy process. If d = 1 and µ(·) is a Sα(0, γ, 0) distribution, then the symbol andLevy triplet of the corresponding stable Levy process are given by p(ξ) = γ|ξ|α and(

0, 0, γα2α−1Γ(α+1

2)

π12 Γ(1− α

2)

dx

|x|α+1

),

respectively.

Let us generalize the notion of one-dimensional symmetric stable Levy processes. Letp : R× R −→ R be given by p(x, ξ) = γ(x)|ξ|α(x), for some functions α : R −→ (0, 2) andγ : R −→ (0,∞). In [43, Theorem 3.3] it is shown that if the functions α(x) and γ(x)satisfy

(i) α, γ ∈ C1b (R) and

(ii) 0 < infx∈R α(x) ≤ supx∈R α(x) < 2 and 0 < infx∈R γ(x) ≤ supx∈R γ(x) <∞,

then the function (symbol) p(x, ξ) = γ(x)|ξ|α(x) defines a Feller process on R with the Levytriplet (

0, 0, γ(x)α(x)2α(x)−1Γ(α(x)+1

2)

π12 Γ(1− α(x)

2)

dy

|y|α(x)+1

)called a stable-like process. Furthermore, by Theorem 2.4.10, the associated semigroupsatisfies the Cb-Feller property and strong Feller property.

As mentioned, since we are only interested in stochastic processes with the same finite-dimensional distributions, we can deal only with canonical cadlag Feller processes (ev-ery sample path is a cadlag function) with stochastic basis which satisfies the usual

2.5 Characteristics of semimartingales 23

conditions. Indeed, let X = ((Rd)R+ , (B(Rd))R+ , Pxx∈Rd ,F0, Xtt∈R+ , θtt∈R+) be acanonical Feller process. Then, the structure (D(Rd), (B(Rd))R+ ∩ D(Rd), PxDx∈Rd ,F0 ∩D(Rd), Xtt∈R+ , θtt∈R+) is a Feller process with the same finite-dimensional distri-butions as the Feller process X, where PxD(· ∩ D(Rd)) := Px(·) for all x ∈ Rd andF0 ∩ D(Rd) := F0

t ∩ D(Rd) = σXs : s ∈ R+, s ≤ tt∈R+ (see [21, Theorem 3.4.24]).Now, by employing the completion as in Theorem 2.4.5 we have the claim.

2.5 Characteristics of semimartingales

Throughout this section we consider only continuous-time stochastic processes, i.e., thetime set is T = R+, and we always assume that the stochastic basis (Ω,F ,P,F) satisfiesthe usual conditions.

Let (Ω,F ,F) be a filtered space. The optional σ-algebra is the σ-algebra O on Ω×R+

that is generated by all cadlag F-adapted processes X : Ω×R+ −→ Rd. A process or a setthat is O-measurable is called optional. The predictable σ-algebra is the σ-algebra Pon Ω×R+ that is generated by all cag F-adapted processes X : Ω×R+ −→ Rd. A processor a set that is P-measurable is called predictable. By [22, Proposition I.1.24], P ⊆ O.

Let (Ω,F ,P) be a probability space. A set A ⊆ Ω×R+ is called evanescent if P(ω ∈Ω : ∃t ∈ R+ with (ω, t) ∈ A) = 0.

Definition 2.5.1 Let (Ω,F ,P,F) be a stochastic basis and let Mtt∈R+ be an R-valuedstochastic process adapted to the filtration F. The process Mtt∈R+ is called a martingaleprovided

(i) E[|Mt|] <∞ for all t ∈ R+

(ii) E[Mt|Fs] = Ms P-a.s. for all s, t ∈ R+, 0 ≤ s ≤ t.

Without loss of generality, we assume that every martingale is a cadlag process (see [35,Theorem II.2.9]). Let us remark that with ≥ (respectively, ≤) in (ii) we obtain a sub-martingale (respectively, supermartingale). The following theorem will be useful.

Theorem 2.5.2 [13, Proposition VII.1.7] Let X be a cadlag universal Markov processwith infinitesimal generator A : DA −→ Bb(Rd). Then

M ft := f(Xt)−

∫ t

0

Af(Xs)ds

is a martingale for all f ∈ DA. 2

Definition 2.5.3 Let (Ω,F ,P,F) be a stochastic basis and let Mtt∈R+ be an R-valuedstochastic process.

(i) The process Mtt∈R+ is called a local martingale if there exists an increasingsequence of stoping times Tnn∈N, such that Tn ↑ ∞ P-a.s. and MTn

t := MTn∧tt∈R+

is a martingale for all n ∈ N.


(ii) The process Mtt∈R+ is called a square-integrable martingale if

supt∈R+

E[|Mt|2] <∞.

(iii) The process Mtt∈R+ is called a locally square-integrable martingale if thereexists an increasing sequence of stoping times Tnn∈N, such that Tn ↑ ∞ P-a.s. andMTn

t t∈R+ is a square-integrable martingale.

Let f : [a, b] −→ R. Define

V (f ; [a, b]) := sup

n∑i=1

|f(xi)− f(xi−1)| : a = x0 < x1 < . . . < xn = b, n ∈ N

.

The number V (f ; [a, b]) is called the total variation of the function f(x) over [a, b]. Iff : [0,∞) −→ R, we say that f(x) has finite variation if V (f ; [0, t]) <∞ for all t ∈ R+.

Definition 2.5.4 Let (Ω,F ,P,F) be a stochastic basis. An F-adapted R-valued cadlagstochastic process Btt∈R+ is said to be of finite variation if the function t 7−→ Bt(ω)has finite variation P-a.s.

Theorem 2.5.5 [22, Theorem I.4.2] For each pair of locally square-integrable martingalesMtt∈R+ and Ntt∈R+ there exists a unique, up to an evanescent set, predictable of finitevariation process 〈M,N〉tt∈R+ , such that (MN − 〈M,N〉)tt∈R+ is a local martingale.Furthermore, if Mtt∈R+ is continuous, then 〈M,M〉tt∈R+ is also continuous. 2

Theorem 2.5.6 [22, Theorem I.4.18] Any local martingale Mtt∈R+ admits a unique,up to an evanescent set, decomposition

Mt = M0 +M ct +Md

t ,

where M c0 = Md

0 = 0, M ct t∈R+ is a continuous local martingale and Md

t t∈R+ is a purelydiscontinuous local martingale, i.e., (MdN)tt∈R+ is a local martingale for every continu-ous local martingale Ntt∈R+ . 2

Definition 2.5.7 Let (Ω,F ,P,F) be a stochastic basis.

(i) A semimartingale is a process Stt∈R+ of the form St = S0+Mt+Bt for all t ∈ R+,where S0 is finite-valued F0/B(R) measurable random variable, Mtt∈R+ is a localmartingale with M0 = 0 and Btt∈R+ is a finite variation process with B0 = 0.

(ii) A special semimartingale is a semimartingale Stt∈R+ which admits the decom-position St = S0 +Mt +Bt as above, with the process Btt∈R+ as predictable.

(iii) Let X = (Ω,F , Pxx∈R,F, Xtt∈R+ , θtt∈R+) be a universal Markov process. Astochastic process Xtt∈R+ is a Markov (special) semimartingale if Xtt∈R+ is


a (special) semimartingale with respect to every Px(·), x ∈ R. Analogously, we definea Feller (special) semimartingale.

Let us remark that the decomposition St = S0 + Mt + Bt of a semimartingale Stt∈R+ isnot in general unique. But there is at most one such decomposition, up to an evanescentset, with Btt∈R+ being predictable (see [22, Corollary I.3.16]). A simple consequence ofTheorem 2.5.6 and [22, Lemma I.4.14] is that there exists a unique, up to an evanescentset, continuous local martingale Sct t∈R+ with Sc0 = 0, such that any decomposition St =S0 + Mt + Bt meets Sct = M c

t (up to an evanescent set). The stochastic process Sct t∈R+

is called the continuous martingale part of Stt∈R+ .For a cadlag stochastic process Xtt∈R+ we define a process ∆Xtt∈R+ by ∆Xt := Xt−

Xt− and ∆X0 := X0. Connection between semimartingales and special semimartingales isgiven in the following theorem.

Theorem 2.5.8 [22, Lemma I.4.24] If a semimartingale Stt∈R+ satisfies |∆St| ≤ Muniformly in t ∈ R+ for some M ≥ 0, then Stt∈R+ is a special semimartingale. More-over, if St = S0 + Mt + Bt is the unique decomposition of Stt∈R+ , then |∆Mt| ≤ 2Mand |∆Bt| ≤M . In particular, if Stt∈R+ is continuous, then Mtt∈R+ and Btt∈R+ arecontinuous as well. 2

Let (Ω,F ,P,F) be a stochastic basis and let Xt = (X1t , . . . , X

dt )t∈R+ be an Rd-valued

F-adapted stochastic process. The process Xtt∈R+ is said to be a (special) semimartingale(respectively, (local) martingale, (locally) square-integrable martingale, of finite variation)if X i

tt∈R+ is a (special) semimartingale (respectively, (local) martingale, (locally) square-integrable martingale, of finite variation) for every 1 ≤ i ≤ d.

Definition 2.5.9 Let (Ω,F ,P,F) be a stochastic basis. A random measure on R+×Rd

is a family µ = µ(ω, dt, dx)ω∈Ω of nonnegative measures on B(R+) × B(Rd) satisfyingµ(ω, 0 × Rd) = 0 for all ω ∈ Ω.

Define Ω := Ω × R+ × Rd with σ-algebras O := O × B(Rd) and P := P × B(Rd).An O/B(R) measurable (respectively, O/B(R) measurable) function is called an optional(respectively, predictable) function. Let µ be a random measure and let W (ω, t, x) bean optional function. Since the function (t, x) 7−→ W (ω, t, x) is B(R+) × B(Rd)/B(R)measurable for every ω ∈ Ω, we can define the integral process W ∗ µtt∈R+ by

W ∗ µt :=

∫[0,t]×RdW (·, s, x)µ(·, ds, dx),

∫[0,t]×Rd |W (·, s, x)|µ(·, ds, dx) <∞

∞, otherwise,(2.26)

Definition 2.5.10 Let (Ω,F ,P,F) be a stochastic basis and let µ be a random measure.

(i) The random measure µ is called optional (respectively, predictable) random mea-sure if the process W ∗ µtt∈R+ is optional (respectively, predictable) for every op-tional (respectively, predictable) function W (ω, t, x).

(ii) An optional random measure µ is called P-σ-finite if there exists a P-measurablepartition Ann∈N of Ω such that each random variable 1An ∗ µ∞ is integrable.


Theorem 2.5.11 [22, Theorem II.1.8] Let (Ω,F ,P,F) be a stochastic basis and let µ bea P-σ-finite random measure. There exists a random measure, called the compensatorof µ and denoted by µp, which is unique up to P-null set, and which is characterized as apredictable random measure satisfying

E[W ∗ µp∞] = E[W ∗ µ∞]

for every nonnegative P/B(R) measurable function W (ω, t, x). 2

Since we will be mostly interested in cadlag processes, the most important example ofa random measure is the measure associated with the jumps of the underlying process.

Theorem 2.5.12 [22, Proposition II.1.16] Let (Ω,F ,P,F) be a stochastic basis and letXtt∈R+ be an Rd-valued F-adapted cadlag process. Then,

µ(·, dt, dx) :=∑s∈R+

1∆Xs 6=0δ(s,∆Xs)(dt, dx)

is an optional and P-σ-finite random measure on R+ × Rd which satisfies

(i) µ(ω, t × Rd) ≤ 1 for all ω ∈ Ω

(ii) for each A ∈ B(R+)× B(Rd), µ(ω,A) takes its values in N ∪ ∞. 2

A bounded and B(Rd)/B(Rd) measurable function h(x) satisfying h(x) = x in a neigh-borhood of the origin is called a truncation function. Let (Ω,F ,P,F) be a stochasticbasis and let Stt∈R+ be an Rd-valued semimartingale. Furthermore, fix a truncationfunction h(x) and define

S(h)t :=∑s≤t

(∆Ss − h(∆Ss)) and S(h)t := St − S(h)t.

Note that above processes are well defined and S(h)tt∈R+ is a finite variation pro-cess (since h(x) is a truncation function and Stt∈R+ is a cadlag process). Therefore,S(h)tt∈R+ is a semimartingale. Since ∆S(h)t = h(∆St), which is uniformly boundedin t ∈ R+, by Theorem 2.5.8, S(h)tt∈R+ is a special semimartingale with the uniquedecomposition

S(h)t = S0 +M(h)t +B(h)t. (2.27)

Definition 2.5.13 Let h : Rd −→ Rd be a truncation function and let Stt∈R+ be asemimartingale.

(a) The triplet (B,C,N) consisting of

(i) the predictable process of finite variation Bt := B(h)t, t ∈ R+, appearing in(2.27)


(ii) the continuous process Cijt := 〈Sc,i, Sc,j〉t for 1 ≤ i, j ≤ d and t ∈ R+

(iii) the compensator of the random measure

µ(·, dt, dx) :=∑s∈R+

1∆Ss 6=0δ(s,∆Ss)(dt, dx),

i.e., a predictable random measure N on R+ × Rd satisfying E[W ∗ N∞] =E[W ∗ µ∞] for every nonnegative P/B(R) measurable function W (ω, t, x)

is called the characteristics of the semimartingale Stt∈R+ (with respect to h(x)).

(b) Let us put Cijt := 〈M(h)j,M(h)j〉t, t ∈ R+, where the process M(h)tt∈R+ appears

in (2.27). Then, the triplet (B, C,N) is called the modified characteristics of thesemimartingale Stt∈R+ (with respect to h(x)).

Let us remark here that the processes Ctt∈R+ and Ctt∈R+ are well defined since theprocesses Sct t∈R+ and M(h)tt∈R+ are locally square integrable. Indeed, by defin-ing a sequence of stopping times T in := inft ∈ R+ : M(h)it > n, n ∈ N, we have

|M(h)T in,it | ≤ n + |∆M(h)

T in,it |. Since ∆S(h)t = h(∆St) and h(x) is bounded, by Theorem

2.5.8, ∆M(h)itt∈R+ is also bounded for all 1 ≤ i ≤ d. In the case of the process Sct t∈R+ ,

because of the continuity, we have ∆Sc,it = 0 for all 1 ≤ i ≤ d, which proves the assertion.Note that only the first two characteristics depend on the underlying truncation function.

Therefore, by replacing the truncation function by another truncation function only thesetwo characteristics will be modified. Transition from the characteristics to the modifiedcharacteristics, and conversely, is given in the following theorem.

Theorem 2.5.14 [22, Proposition II.2.17] Up to an evanescent set,

Cijt = Cij

t + (hihj) ∗Nt −∑s≤t

∆Bis∆B

js

for 1 ≤ i, j ≤ d. 2

The following characterization of the characteristics will be useful in the following chap-ters.

Theorem 2.5.15 [22, Theorem II.2.42] A semimartingale Stt∈R+ admits the charac-teristics (B,C,N) (with respect to a truncation function h(x)) if, and only if, for eachf ∈ C2

b (Rd) the process

f(St)− f(S0)−∑

1≤i≤d

∫ t

0

∂

∂xif(Ss−)dBi

s −1

2

∑1≤i,j≤d

∫ t

0

∂2

∂xi∂xjf(Ss−)dCij

s

−∫ t

0

∫Rd

(f(Ss− + x)− f(Ss−)−

∑1≤i≤d

∂

∂xif(Ss−)hi(x)

)N(·, ds, dx)


is a local martingale. 2

Connection between Feller processes and semimartingales is given in the following the-orem.

Theorem 2.5.16 [42, Lemma 3.2 and Theorem 3.5] Let X be an Rd-valued Feller processwith infinitesimal generator A : DA −→ C0(Rd), satisfying C∞c (Rd) ⊆ DA, and symbolp(x, ξ) given by (2.25). Assume that ||p(·, ξ)||∞ ≤ c(1 + |ξ|2) for all ξ ∈ Rd, for somec ≥ 0, and p(x, 0) = 0. Then, X is a Feller semimartingale. Moreover, if h(x) := xχR(x),where χR ∈ C∞c (Rd) such that 1y∈Rd:|y|≤R(x) ≤ χR(x) ≤ 1y∈Rd:|y|≤2R(x), and (b, c, ν)is the Levy triplet of p(x, ξ) (with respect to the function h(x) = 1y:|y|≤1(x)), then thesemimartingale characteristics (B,C,N) of X is given by

Bit =

∫ t

0

bi(Xs)ds+

∫ t

0

∫Rd

(yiχR(y)− yi1z:|z|≤1(y)

)ν(Xs, dy)ds,

Cijt =

∫ t

0

cij(Xs)ds and

N(·, dt, dx) = dtν(Xs, dx)

for all 1 ≤ i, j ≤ d. 2

From the above theorem we read that every Levy process is a semimartingale with thecharacteristics

Bit = tbi + t

∫Rd

(yiχR(y)− yi1z:|z|≤1(y)

)ν(dy),

Cijt = tcij and

N(dt, dx) = dtν(dx)

for all 1 ≤ i, j ≤ d.

Definition 2.5.17 A semimartingale Stt∈R+ is called a homogeneous diffusion withjumps if its (modified) characteristics have the form

Bit =

∫ t

0

bi(Ss)ds,

Cijt =

∫ t

0

cij(Ss)ds,

N(·, dt, dx) = dtν(Xs, dx) and

Cijt =

∫ t

0

cij(Ss)ds =

∫ t

0

cij(Ss)ds+

∫ t

0

∫Rdhi(x)hj(x)ν(Ss, dx)ds,

where the functions bi(x) and cij(x), i, j = 1, . . . , d, are B(Rd)/B(R) measurable, c(x) :=(cij(x))di,j=1 is a symmetric nonnegative-definite d× d matrix-valued function and ν(x, ·) is

2.6 Convergence of semimartingales 29

a nonnegative Borel kernel on Rd × B(Rd) such that for every x ∈ Rd, ν(x, 0) = 0 and∫Rd(1 ∧ |y|

2)ν(x, dy) <∞.

Clearly, every Feller process which satisfies the conditions from Theorem 2.5.16 is a homo-geneous diffusion with jumps.

2.6 Convergence of semimartingales

Throughout this chapter S will denote a Polish space, i.e., a complete separable metricspace, and B(S) will denote the Borel σ-algebra on S. We say that a sequence of probabilitymeasures Pnn∈N on (S,B(S)) converges weakly to the probability measure P(·), andwrite Pn ⇒ P, if ∫

S

fdPn −→∫S

fdP as n −→∞

for all f ∈ Cb(S). A set B ∈ B(S) whose boundary ∂B satisfies P(∂B) = 0 is called aP-continuity set. In the following theorem the weak convergence is characterized.

Theorem 2.6.1 [4, Theorem 2.1] Let Pnn∈N and P(·) be probability measures on(S,B(S)). The following conditions are equivalent

(i) Pn ⇒ P

(ii)∫SfdPn −→

∫SfdP as n −→∞ for all bounded uniformly continuous functions f(s)

(iii) lim supn−→∞ Pn(F ) ≤ P(F ) for all closed sets F ⊆ S

(iv) lim infn−→∞ Pn(O) ≥ P(O) for all open sets O ⊆ S

(v) limn−→∞ Pn(B) = P(B) for all P-continuity sets B ⊆ S. 2

In many situations the above characterizations are not very useful. In order to get, insome situations, more operable characterization of the weak convergence we introduce thenotion of tightness. A family of probability measures Π on (S,B(S)) is said to be tight iffor every ε > 0 there exists a compact set C ⊆ S, such that P(C) > 1− ε for all P ∈ Π.

Let Π be a family of probability measures on (S,B(S)). We call the family Π relativelycompact if every sequence of elements in Π has a weakly convergent subsequence. Letus remark that if the space of all probability measures on (S,B(S)), denoted by P(S),is topologized with the weak topology, then it becomes a Polish space and its subset isrelatively compact if, and only if, its closure is compact.

Theorem 2.6.2 [4, Theorems 5.1 and 5.2] A family of probability measures Π on (S,B(S))is relatively compact if, and only if, it is tight. 2

As a consequence of the previous theorem we get the following. If a sequence of prob-ability measures Pnn∈N on (S,B(S)) is tight and if every subsequence that converges at


all in fact converges to a probability measure P(·), then Pn ⇒ P. Indeed, since a necessaryand sufficient condition for Pn ⇒ P is that each subsequence Pnii∈N of Pnn∈N containsa further subsequence Pnij j∈N converging weakly to P, we have the claim.

We are mostly interested in convergence of stochastic processes, i.e., random elements.Let (Ωn,Fn,Pn)n∈N and (Ω,F ,P) be probability spaces and let Xn and X be Fn/B(S),n ∈ N, and F/B(S) measurable random elements, respectively. We say that the sequenceof random elements Xnn∈N converges in distribution to the random element X, andwrite Xn ⇒ X, if PnXn ⇒ PX .

Since the stochastic processes we deal with are cadlag processes, we want to topologizethe space D(Rd) up to a Polish space. This can be done by introducing the so-calledSkorokhod metric

δ(α, β) =∞∑n=1

2−n(1 ∧ δn(α, β))

for α, β ∈ D(Rd), where

δn(α, β) = infλ∈Λ

(|||λ|||+ ||(knα) λ− knβ||∞),

Λ is the set of all strictly increasing continuous functions λ : R+ −→ R+ with λ(0) = 0and λ(t) ↑ ∞ as t −→∞,

|||λ||| = sups<t

∣∣∣∣logλ(t)− λ(s)

t− s

∣∣∣∣ and kn(t) =

1, t ≤ n

n+ 1− t, n < t < n+ 10, t ≥ n+ 1.

Furthermore, if we denote by D(Rd) = σπt : t ∈ R+ the σ-algebra generated by projec-tions πt : D(Rd) −→ Rd, πt(α) := α(t), for t ∈ R+, then D(Rd) = B(D(Rd)). For detailsand proofs of the stated results see [22, Chapter VI].

Hence, in order to prove the convergence in distribution of a sequence of Rd-valuedcadlag stochastic processes Xn = Xn

t t∈R+ defined on probability spaces (Ωn,Fn,Pn),n ∈ N, to an Rd-valued cadlag stochastic process X = Xtt∈R+ defined on a probabilityspace (Ω,F ,P), it suffices to prove that the sequence of probability measures PnXnn∈N istight and every subsequence that converges at all in fact converges to a probability measurePX(·). Clearly, we can assume that all the processes Xn, n ∈ N, and X are defined onthe same probability space. Otherwise, we just take the product of the correspondingprobability spaces. In the case when the processes Xn, n ∈ N, and X are semimartingales,sufficient conditions for Xn ⇒ X are given in the following theorems.

Theorem 2.6.3 [22, Theorem VIII.2.17] Let Sn = Snt t∈R+ , n ∈ N, be a sequence of

Rd-valued semimartingales with the modified characteristics (Bn, Cn, Nn)n∈N, and let S =Stt∈R+ be an Rd-valued stochastic process with stationary and independent increments,

i.e., a Levy process, and the modified characteristics (B, C,N). Further, without loss ofgenerality, assume that Sn, n ∈ N, and S are defined on a stochastic basis (Ω,F ,P,F).

2.6 Convergence of semimartingales 31

Then, Sn ⇒ S if

Sn0 ⇒ S0, sups≤t|Bn

s −Bs|P−→ 0, Cn

tP−→ Ct and g ∗Nn

tP−→ g ∗Nt

for all g ∈ Cb(Rd) vanishing in a neighborhood of the origin and all t ∈ R+, whereP−→

denotes the convergence in probability. 2

Theorem 2.6.4 [22, Theorem IX.4.8] Let h(x) be an arbitrary truncation function andlet Sn = Snt t∈R+ , n ∈ N, be a sequence of Rd-valued homogeneous diffusions with jumpsdefined on stochastic basis (Ω,F ,P,F) with the modified characteristics (with respect tothe truncation function h(x))

Bn,it =

∫ t

0

bn,i(Ss)ds, Cn,ijt =

∫ t

0

cn,ij(Ss)ds and Nn(·, dt, dx) = dtνn(Ss, dx),

for 1 ≤ i, j ≤ d and n ∈ N. Furthermore, let S = Stt∈R+ be a canonical stochasticprocess on the filtered space (D(Rd),D(Rd),D(Rd)), where the filtration D(Rd) is definedas D(Rd) = Dt(Rd)t∈R, Dt(Rd) :=

⋂s>tD0

s(Rd) and D0t (Rd) := Ss : s ∈ R+, s ≤ t for

t ∈ R+. Assume that

(i) b(x) is a B(Rd)/B(Rd) measurable function, c(x) is a symmetric nonnegative-definited× d matrix-valued B(Rd)/B(Rd+d) measurable function and ν(x, ·) is a nonnegativeBorel kernel on Rd × B(Rd) such that for every x ∈ Rd, ν(x, 0) = 0 and

∫Rd(1 ∧

|y|2)ν(x, dy) <∞

(ii) the functions b(x),

c(x) :=

cij(x) +

∫Rdhi(y)hj(y)ν(x, dy)

1≤i,j≤d

and x 7−→∫Rd g(y)ν(x, dy) are continuous for all g ∈ Cb(Rd) vanishing in a neighbor-

hood of the origin

(iii) bn −→ b, cn −→ c and∫Rd g(y)νn(x, dy) −→

∫Rd g(y)ν(x, dy) converge locally uni-

formly for all g ∈ Cb(Rd) vanishing in a neighborhood of the origin

(iv) for all a > 0,limb−→∞

sup|x|≤a

ν(x, y : |y| > b) = 0

(iv) PSn0 ⇒ µ, where µ(·) is a distribution on B(Rd)

(v)

Bit :=

∫ t

0

bi(Ss)ds, N(·, dt, dx) := dtν(Ss, dx) and Cijt :=

∫ t

0

cij(Ss)ds,


for 1 ≤ i, j ≤ d

(vi) for each x ∈ Rd there exists a probability measure Qx(·) on (D(Rd),D(Rd)) (whichis actually unique (see [22, Theorem III.2.16])), such that Qx

S0(·) = δx(·) and S is a

homogeneous diffusion with jumps on the stochastic basis (D(Rd),D(Rd),Qx,D(Rd))with the modified characteristics (B, C,N) (with respect to the truncation functionh(x))

(vii) x 7−→ Qx(Γ) is B(Rd)/B(R) measurable for all Γ ∈ D(Rd).

Then, PSn ⇒ Qµ, where Qµ(·) :=∫Rd Q

x(·)µ(dx), and S is a homogeneous diffusion withjumps on the stochastic basis (D(Rd),D(Rd),Qµ,D(Rd)) with the modified characteristics(B, C,N) (with respect to the truncation function h(x)). 2

Let us remark that, since we are interested in convergence of semimartingales, in theabove theorems we choose the truncation functions to be continuous. Finally, we give anapproximation of Feller processes by Markov chains.

Theorem 2.6.5 [8, Theorem 1] Let X be an Rd-valued Feller process with infinitesimalgenerator A : DA −→ C0(Rd), such that C∞c (Rd) is an operator core of A, i.e., (A,DA)is the only extension of (A|C∞c (Rd), C

∞c (Rd)) generating a Feller semigroup. Furthermore,

let p(x, ξ) be the symbol of A|C∞c (Rd), given by (2.25), such that ||p(·, ξ)||∞ ≤ c(1 + |ξ|2),for some c ≥ 0, and p(x, 0) = 0. For each m ∈ N define a Markov chain Xm with Xm

0 = X0

and transition function pm(x,B) for x ∈ Rd and B ∈ B(Rd), such that∫Rdei〈ξ,y〉pm(x, dy) = ei〈ξ,x〉−

1mp(x,ξ)

for all m ∈ N. Then Xm ⇒ X as m −→∞, where X

m:= Xm

bmtct∈R+ . 2

2.7 Recurrence and transience of Markov models

Throughout this section X = (Ω,F , Pxx∈Rd ,F, Xtt∈T, θtt∈T) will denote an Rd-valued (cadlag) strong universal Markov model. For x ∈ Rd and B ∈ B(Rd), define ηB :=∫T 1Xt∈Bµ(dt), τB := inft ∈ T : Xt ∈ B, U(x,B) := Ex(ηB), Q(x,B) := Px(ηB = ∞)

and L(x,B) := Px(τB <∞). Here, µ(·) denotes the counting measure in the discrete-timecase and the Lebesgue measure in the continuous-time case.

Definition 2.7.1 A Markov model X is ϕ-irreducible if there exists a σ-finite measureϕ(·) on B(Rd), such that whenever ϕ(B) > 0, we have U(x,B) > 0 for all x ∈ Rd.

Let us remark that in the discrete-time case we have equivalent and more operable definitionof irreducibility. A Markov chain X is ϕ-irreducible if there exists a σ-finite measure ϕ(·)on B(Rd), such that for every x ∈ Rd and every B ∈ B(Rd) with ϕ(B) > 0, there existsn ∈ N such that pn(x,B) > 0.

2.7 Recurrence and transience of Markov models 33

In [46, Theorem 2.1] it is shown that the irreducibility measure can always be maximized.If X is a ϕ-irreducible Markov model, then there exists a σ-finite measure ψ(·) on B(Rd)such that X is ψ-irreducible and ϕ′ ψ, for every irreducibility measure ϕ′(·) on B(Rd) ofX. The measure ψ(·) is called the maximal irreducibility measure and from now on,when we refer to an irreducibility measure we actually refer to the maximal irreducibilitymeasure. For a ψ-irreducible Rd-valued Markov model X, set B+(Rd) = B ∈ B(Rd) :ψ(B) > 0.

Definition 2.7.2 Let X be a Markov model.

(i) A set B ∈ B(Rd) is uniformly transient if there exists a finite constant M ≥ 0such that U(x,B) ≤M holds for all x ∈ Rd.

(ii) A set B ∈ B(Rd) is transient if it can be covered by countable number of uniformlytransient sets.

(iii) A set B ∈ B(Rd) is recurrent if U(x,B) =∞ holds for all x ∈ Rd.

(iv) A set B ∈ B(Rd) is Harris recurrent, or H-recurrent, if Q(x,B) = 1 holds for allx ∈ Rd.

(v) The model X is transient if it is ψ-irreducible and if Rd is transient.

(vi) The model X is recurrent if it is ψ-irreducible and if every set B ∈ B+(Rd) isrecurrent.

(vii) The model X is H-recurrent if it is ψ-irreducible and if every set B ∈ B+(Rd) isH-recurrent.

In [29, Proposition 9.1.1] and [28, Theorem 2.4] it is shown that a set B ∈ B+(Rd) is aH-recurrent set if, and only if, L(x,B) = 1 holds for all x ∈ Rd. Hence, the H-recurrenceproperty can be also defined in the following way. A Markov model X is H-recurrent if it isψ-irreducible and if L(x,B) = 1 holds for all x ∈ Rd and all B ∈ B+(Rd). The well-knowndichotomy between recurrence and transience is given in the following theorem.

Theorem 2.7.3 [46, Theorem 2.3] Every ψ-irreducible Markov model is either recurrentor transient. 2

Clearly, H-recurrence implies recurrence, but in general these two properties are notequivalent. They differ on a set of the irreducibility measure zero.

Definition 2.7.4 Let X be a ψ-irreducible Markov model.

(i) A set B ∈ B(Rd) is called an absorbing set, if pt(x,B) = 1 for all x ∈ B and allt ∈ T.

(ii) A set D ∈ B(Rd) is called a maximal absorbing set, if D = x ∈ Rd : Q(x,D) =1.


(iii) A set H ∈ B(Rd) is called a Harris maximal set, if the set H is a maximal absorbingset and if ψ(B) > 0 implies Q(x,B) = 1 for all x ∈ H.

Note that any maximal absorbing set is an absorbing set. Indeed, by the Markov propertywe have

Px(∫

s≥t1Xs∈Dµ(ds) =∞

)= Ex

[Ex[1ηH=∞ θt|Ft]

]= Ex

[EXt [1ηH=∞]

]=

∫D

Q(y,D)pt(x, dy) +

∫DcQ(y,D)pt(x, dy)

= pt(x,D) +

∫DcQ(y,D)pt(x, dy).

Hence, if there would exist x0 ∈ D and t0 ∈ T such that pt0(x0, D) < 1, then we would have

Px0(∫

s≥t0 1Xs∈Dµ(ds) =∞)< 1, which is impossible. Therefore, the above definition

actually says that the restriction of a Markov model to a Harris maximal set is H-recurrentMarkov model. But this is not completely true, since in the continuous-time case it is notclear if we can conclude that Px(Xt ∈ H for all t ∈ R+) = 1 holds true, unless H is a closedset.

Theorem 2.7.5 [46, Theorem 2.5] Let X be a ψ-irreducible recurrent Markov model.Then, Rd = H ∪Hc, where H ∈ B+(Rd) is a Harris maximal set with ψ(Hc) = 0 and Hc

is a transient set. 2

Note that for a Markov model on a countable state space, since the irreducibility measureis the counting measure, the recurrence and H-recurrence properties are equivalent.

Since we are mostly interested in Markov models on the state space Rd, which has veryrich topological structure, in the sequel we will employ certain topological properties ofRd. Recall that a function f : R −→ R is called a lower semicontinuous function iflim infy−→x f(y) ≥ f(x) holds for all x ∈ R.

Definition 2.7.6 Let X be a Markov model.

(i) A set C ∈ B(Rd) is called a νa-petite set if there exist a distribution a(·) on T anda nontrivial measure νa(·) on B(Rd) such that∫

Tpt(x,B)a(dt) ≥ νa(B)

holds for all x ∈ C and all B ∈ B(Rd).

(ii) The model X is called a T-model if for some distribution a(·) on T there exists akernel T (x, ·) on Rd×B(Rd) with T (x,Rd) > 0 for all x ∈ Rd, such that the functionx 7−→ T (x,B) is lower semicontinuous for all B ∈ B(Rd), and∫

Tpt(x,B)a(dt) ≥ T (x,B)


holds for all x ∈ Rd and all B ∈ B(Rd).

(iii) A state x ∈ Rd is called a topologically recurrent state if U(x,Ox) = ∞, forevery open set Ox containing x. Otherwise, x is called a topologically transientstate.

Petite sets can be understood as sets which take place of singletons, and for Markovmodels on countable state spaces each singleton is a petite set.

Theorem 2.7.7 [29, Theorems 8.3.4 and 8.3.5 and Proposition 9.1.7] [28, Theorem 3.3]Let X be a ψ-irreducible Markov model.

(i) The model X is H-recurrent if, and only if, there exists a H-recurrent petite set.

(ii) In the continuous-time case, assume that for every petite set C ∈ B(Rd) there existsa distribution aC(·) on R+, such that

infx∈C

∫ ∞0

Px(Xt ∈ B)aC(dt) > 0 (2.28)

holds for all B ∈ B+(Rd). Then, the model X is transient if, and only if, every petiteset is uniformly transient.

(iii) By assuming the relation in (2.28) for the continuous-time case, the model X isrecurrent if, and only if, there exists a recurrent petite set.

Proof.

(i) The proof is given in [29, Proposition 9.1.7] and [28, Theorem 3.3].

(ii) The proof for the discrete-time case is given in [29, Theorems 8.3.5]. To prove theclaim for the continuous-time case, we proceed as follows. If the model X is transient,then there exists at least one uniformly transient set B ∈ B+(Rd). Due to the relationin (2.28),

δC := infx∈C

∫ ∞0

Px(Xt ∈ B)aC(dt) > 0

holds for every petite set C ∈ B(Rd). Using the Chapman-Kolmogorov equation wehave

U(x,B) =

∫ ∞0

U(x,B)aC(dt) =

∫ ∞0

∫ ∞0

ps(x,B)dsaC(dt)

≥∫ ∞

0

∫ ∞0

ps+t(x,B)dsaC(dt) =

∫ ∞0

∫ ∞0

∫Rdps(x, dy)pt(y,B)dsaC(dt)

≥ δC

∫ ∞0

∫C

ps(x, dy)ds = δCU(x,C).

Hence, the claim follows.


(iii) The proof follows directly from (ii). 2

Recall that the support of a positive measure µ(·) on B(Rd), denoted by suppµ, is theset of all points x ∈ Rd such that µ(Ox) > 0 for all open neighborhoods Ox of x. Clearly,suppµ is always a closed set. By assuming certain continuity properties of Markov models,we can conclude that every compact set is a petite set.

Theorem 2.7.8 [46, Theorems 5.1 and 7.1] [29, Proposition 6.1.1]

(i) A ψ-irreducible Markov model is a T-model if, and only, if every compact set is apetite set.

(ii) If X is a ψ-irreducible Markov model such that its corresponding semigroup satisfiesthe Cb-Feller property and suppψ has nonempty interior, then X is a T-model.

(iii) The semigroup of a Markov model X satisfies the Cb-Feller property if, and only if,the function x 7−→ pt(x,O) is lower semicontinuous for all t ∈ T and all open setsO ⊆ Rd.

(iv) The semigroup of a Markov model X satisfies the strong Feller property if, and onlyif, the function x 7−→ pt(x,B) is lower semicontinuous for all t ∈ T \ 0 and allB ∈ B(Rd). 2

Involving topological properties of Markov models from the previous theorem we canrewrite Theorem 2.7.5 as follows.

Theorem 2.7.9 [46, Theorem 4.2] Let X be a ψ-irreducible T-model. Then, Rd = H∪Hc,where the set H is either empty or Harris maximal set and Hc is a transient set withψ(Hc) = 0, and every state in Hc is topologically transient state. 2

In the following chapters we will be mostly interested in processes with jump distribu-tions which are absolutely continuous with respect to the Lebesgue measure. Hence, let usassume that X is a λ-irreducible Markov model.

Proposition 2.7.10 Let X be a λ-irreducible T-model. Then, X is either H-recurrent ortransient.

Proof. By Theorem 2.7.3, it suffices to prove that the recurrence of X implies its H-recurrence. Let x ∈ Rd be arbitrary and let Ox be an arbitrary open neighborhood of x.Then, since λ(Ox) > 0, by the recurrence we have U(x,Ox) = ∞, i.e., x is topologicallyrecurrent state. Since x ∈ Rd is arbitrary, by Theorem 2.7.9, the state space Rd is a Harrismaximal set, i.e., X is H-recurrent. 2

Proposition 2.7.11 Let X and Y be λ-irreducible T-models. If there exists a compact

set C ⊆ Rd, with λ(C) > 0, such that τXCd= τYC for every starting point x ∈ Cc, then the

model X is recurrent if, and only if, the model Y is recurrent.


Proof. By assumption, LX(x,C) = LY(x,C) holds for all x ∈ Cc, and the equality on theset C is trivially satisfied. The claim now follows from Theorem 2.7.7 (i) and (ii), Theorem2.7.8 (i) and Proposition 2.7.10. 2

Let B ∈ B(Rd) be arbitrary. In the continuous-time case, define the set of recurrentpaths by

R(B) := ω ∈ D(Rd) : ∀n ∈ N, ∃t ≥ n such that ω(t) ∈ B,

and the set of transient paths by

T (B) := ω ∈ D(Rd) : ∃s ≥ 0 such that ω(t) 6∈ B, ∀t ≥ s.

It is clear that T (B) = R(B)c, and for any open set O ⊂ Rd, by the right continuity, R(O)and T (O) are D(Rd) measurable. In the discrete-time case, using the same notation, wesimilarly define the set of recurrent paths by

R(B) := ω ∈ (Rd)Z+ : ∀n ∈ N, ∃m ≥ n such that ω(m) ∈ B,

and the set of transient paths by

T (B) := ω ∈ (Rd)Z+ : ∃m ≥ 0 such that ω(n) 6∈ B, ∀n ≥ m.

Clearly, T (B) = R(B)c and for any B ∈ B(Rd), R(B) and T (B) are (B(R))Z+ measurable.

Proposition 2.7.12 Let X be a λ-irreducible T-model, and let us assume (2.28) for thecontinuous-time case. Then, the following 0-1 property must be met

Px (ηO =∞) = 0 for all x ∈ Rd and all open bounded sets O ⊆ Rd

orPx (ηO =∞) = 1 for all x ∈ Rd and all open bounded sets O ⊆ Rd.

In particular, the model X is recurrent if, and only if, PxX(R(O)) = 1 for all x ∈ Rd and allopen bounded sets O ⊆ Rd, and it is transient if, and only if, PxX(T (O)) = 1 for all x ∈ Rd

and all open bounded sets O ⊆ Rd.

Proof. The 0-1 property follows from Theorems 2.7.7 and 2.7.8 (i) and Proposition 2.7.10,and the characterization by sets of paths easily follows from the 0-1 property and Theorem2.7.7 (i). 2

In the special case when a Markov model X has stationary and independent increments,i.e., when it is a random walk or a Levy process, the notion of recurrence and transiencecan be refined. More precisely, the concept of irreducibility will not be crucial any more.Thus, in this situation, we will be able to study the recurrence and transience of modelswhich are not necessarily irreducible, such as a random walk with the jump distributionδ1(·) or the Poisson process. But, in addition, if the model X is a T-model and there existsa point x ∈ Rd such that L(y,Ox) > 0 holds for all y ∈ Rd and all open neighborhoodsOx around x, then X will be a ψ-irreducible model with ψ(·) := T (x, ·), where the kernel


T (x, ·) is defined in Definition 2.7.6 (see [46, Theorem 3.2]). Every Markov model X withstationary and independent increments satisfies either

U(x,Ox) =∞ (2.29)

for all x ∈ Rd and all open bounded neighborhoods Ox around x, or

U(x,Ox) <∞ (2.30)

for all x ∈ Rd and all open bounded neighborhoods Ox around x (see [40, Theorems35.3 and 35.4]). The relations in (2.29) and (2.30) are also called the recurrence andtransience property, respectively. Even without the irreducibility notion, we have thedichotomy between the recurrence and transience. Further, note that if the model Xis λ-irreducible, then, since its corresponding semigroup satisfies the Cb-Feller property,Theorem 2.7.8 (ii) implies that it is a T-model. Hence, by [46, Theorems 2.2 and 4.1], thisnotion of recurrence and transience coincide with the notion introduced in Definition 2.7.2.

Theorem 2.7.13 [12, Theorem 4.2.9] [40, Theorems 35.3 and 35.4 and Remark 37.7]Let X be a Markov model with stationary and independent increments determined by acharacteristic function ϕ(ξ) (in the random walk case) and symbol p(ξ) (in the Levy processcase), then

(i) the model X is recurrent if, and only if,

lim inft−→∞

|Xt − x| = 0 Px-a.s.

(ii) the model X is transient if, and only if,

limt−→∞

|Xt| =∞ Px-a.s.

(iii) the random walk X is recurrent if, and only if,∫(−δ,δ)d

Re

(1

1− ϕ(ξ)

)dξ =∞

for some δ > 0. Moreover, if the jump distribution of the random walk X is infinitelydivisible with characteristic exponent ψ(ξ), then X is recurrent if, and only if,∫

(−δ,δ)dRe

(1

ψ(ξ)

)dξ =∞

for some δ > 0.

(iv) the Levy process X is recurrent if, and only if,∫(−δ,δ)d

Re

(1

p(ξ)

)dξ =∞


for some δ > 0. 2

Recall that an Rd-valued Markov model X with stationary and independent increments issaid to be truly d-dimensional if it does not live on a proper linear subspace of Rd, i.e.,if for every x ∈ Rd \ 0 there exists t ∈ T, such that

P0(〈Xt, x〉 = 0) < 1

(see [12, page 170] and [40, Definition 24.18]). As a consequence of the previous theoremwe get.

Corollary 2.7.14 [40, Theorems 35.4, 37.8 and 37.14 and Corollary 37.17]

(i) Every truly d-dimensional, d ≥ 3, random walk or Levy process is transient.

(ii) In the dimension d = 2, in the class of truly two-dimensional stable random walks orLevy processes, the only recurrent case is a random walk with zero mean Gaussianjumps or the Brownian motion.

(iii) In the dimension d = 1, in the class of stable random walks or Levy processes withparameters (α, β, γ, δ), the only recurrent cases are

(a) α = 2 and δ = 0

(b) 1 < α < 2 and δ = 0

(c) α = 1 and β = 0. 2

From the previous corollary we easily read that any SαS random walk or Levy process isrecurrent if, and only if, α ≥ 1.

At the end, we consider the discrete-time case only. In general, it is not always easy todetermine the class of petite sets, hence we introduce a more operable class of sets.

Definition 2.7.15 Let X be a Markov chain. A set S ∈ B(Rd) is called a νn-small setif there exist n ∈ N and a nontrivial measure νn(·) on B(Rd), such that for all B ∈ B(Rd)and all x ∈ S we have

pn(x,B) ≥ νn(B).

Clearly, if S is a νn-small set, then S is a νδn-petite set. But, in general, these two notionsare not equivalent. The reason is because small sets deal with one single skeleton only, i.e.,with a Markov chain with pn(x,B) as one-step transition function, and petite sets takeinto account all skeletons. But this property of small sets can be used to define periodicityof Markov chains.

Theorem 2.7.16 [29, Theorem 5.2.2 and Proposition 5.2.4] Let X be a ψ-irreducibleMarkov chain.

(i) For every B ∈ B+(Rd) there exists a νn-small set S ⊆ B, such that S ∈ B+(Rd) andνn(B) > 0.


(ii) If S ∈ B(Rd) is a νn-small set, and if for T ∈ B(Rd) and some m ∈ N we haveinfx∈T pm(x, S) = δ > 0, then T is a νn+m-small set, where νm+n(·) = δνn(·). 2

According to the previous theorem, without loss of generality, fix a νm-small set S ∈B+(R) such that νm(S) > 0. Next, define the following set

ES := n ∈ N : the set S is νn-small set, where νn = δnνm for some δn > 0.

Note that the set ES is closed under addition. Indeed, let i, j ∈ ES, B ∈ B(Rd) and x ∈ Sbe arbitrary. Then, we have

pi+j(x,B) ≥∫S

pi(x, dy)pj(y,B) ≥ (δiδjνm(S))νm(B).

Theorem 2.7.17 [29, Theorem 5.4.4] Let X be a ψ-irreducible Markov chain and letS ∈ B+(Rd) be a νm-small set with νm(S) > 0. Furthermore, let d be the greatest commondivisor of the set ES. Then, there exist disjoint sets D1, . . . , Dd ∈ B(Rd), called a d-cycle,such that

(i) for x ∈ Di, p(x,Di+1) = 1 for all i = 0, . . . , d− 1 mod d

(ii) the set (∪ni=1Di)c is a ψ-null set. 2

Let us remark that the above construction is ψ -a.e. independent of the starting small set,i.e., every two such partitions of Rd coincide ψ -a.e and every small set is ψ -a.e. contained inonly one member of any partition. Moreover, the d-cycle D1 . . . , Dd is maximal in the sensethat for any other partition D′1 . . . , D

′d′ satisfying (i) and (ii), we have d′ dividing d, whilst

if d = d′, then, by reordering the indices if necessary, D′i = Di ψ-a.e. for i = 0, . . . , d− 1.

Definition 2.7.18 Let X be a ψ-irreducible Markov chain.

(i) The largest d ∈ N for which a d-cycle occurs for the chain X is called the period ofX.

(ii) When d = 1, the chain X is called aperiodic.

As we could expect, we have the following.

Theorem 2.7.19 [29, Theorem 5.5.7] Let X be a ψ-irreducible aperiodic Markov chain.Then, every petite set is a small set. 2

The question of recurrence and transience is a very complex problem, and it is practicallyimpossible to give an answer based just on the definition. Therefore, we recall a verypowerful and more operable tool, called the Foster-Lyapunov drift conditions, whichgive necessary and sufficient conditions for recurrence and transience. Basically, the Foster-Lyapunov drift conditions measure the average one-step drifting from the center of the state


space. If this drifting decreases, then the chain is recurrent, and if increases, then the chainis transient.

Let X be a Markov chain and let V (x) be an arbitrary B(Rd)/B(R) measurable function.A drift operator ∆ is defined by the formula

∆V (x) :=

∫Rdp(x, dy)V (y)− V (x).

For a given function V (x) and r ∈ R let

CV (r) := x : V (x) ≤ r.

Theorem 2.7.20 [29, Theorem 8.4.2] Let X be a ψ-irreducible Markov chain. The chainX is transient if, and only if, there exist a bounded and B(Rd)/B(R+) measurable functionV (x) and r ≥ 0 such that

(i) CV (r) and CcV (r) are both in B+(Rd)

(ii) for every x ∈ CcV (r) we have ∆V (x) ≥ 0. 2

Definition 2.7.21 Let X be a Markov chain. A B(Rd)/B(R+) measurable function V (x)is called unbounded off petite sets function for the chain X, if for every r ∈ R the setCV (r) is a petite set.

Theorem 2.7.22 [29, Theorem 8.4.3] Let X be a ψ-irreducible Markov chain. If thereexist a petite set C ⊆ Rd and unbounded off petite sets function V (x) which satisfies∆V (x) ≤ 0 for all x ∈ Cc, then the chain X is H-recurrent. 2

If, in addition, we assume that the semigroup of the underlying Markov chain satisfies theCb-Feller property, then the previous theorem gives necessary and sufficient conditions forH-recurrence (see [29, Theorem 9.4.1]).

Chapter 3

Stable-like Markov chains

In this chapter, we introduce the notion of stable-like Markov chains and discuss certainstructural properties of these Markov chains.

3.1 General stable-like Markov chains

We generalize the notion of a one-dimensional SαS random walk determined by a jumpdistribution with characteristic exponent ψ(ξ) = γ|ξ|α, α ∈ (0, 2), in the way that theindex of stability of jump distribution depends on the current position. In other words,we are interested in Markov chains with transition jumps with characteristic exponentsof the form ψ(x; ξ) = γ(x)|ξ|α(x), where α : R −→ (0, 2) and γ : R −→ (0,∞) are somefunctions. Recall that, by Theorem 2.2.4 (iii), (iv) and (v), the stable density functionsf(y;α(x), γ(x)) have the property

f(y;α(x), γ(x)) ∼ c(x)|y|−α(x)−1,

when |y| −→ ∞ for all x ∈ R, where

c(x) =

γ(x)

2, α(x) = 1

γ(x)π

Γ(α(x) + 1) sin(πα(x)

2

), α(x) 6= 1.

Let us formalize this idea. Let α : R −→ (0, 2) and c : R −→ (0,∞) be arbitraryfunctions and let fxx∈R be a family of probability densities on R satisfying

(C1) x 7−→ fx(y) is B(R)/B(R) measurable function for all y ∈ R

(C2) fx(y) ∼ c(x)|y|−α(x)−1, for |y| −→ ∞.

The functions α(x) and c(x) are called the stability function and scaling function,respectively. We define a Markov chain X on R by the following transition function

p(x, dy) := fx(y − x)dy.

43

44 3. Stable-like Markov chains

Transition densities of the chain X are asymptotically equivalent to the densities of SαSdistributions. Note that when fx(y) = f(y;α, γ) for all x ∈ R, then X is a SαS randomwalk.

Clearly, if suppfx := y ∈ R : fx (y) 6= 0 = R for all x ∈ R, then the chain X isλ-irreducible. Moreover, since the transition function p(x, ·) is equivalent, in the absolutecontinuous sense, to the Lebesgue measure, λ(·) is the maximal irreducibility measure for X(see Proposition 3.1.1). Of course, in general, this does not have to be true. We introducethe following additional conditions

(C3) there exists k > 0 such that

lim|y|−→∞

supx∈[−k,k]c

∣∣∣∣fx(y)|y|α(x)+1

c(x)− 1

∣∣∣∣ = 0

(C4) infx∈C

c(x) > 0 for every compact set C ⊆ [−k, k]c.

Condition (C3) ensures that out of some compact set, all jump densities can be replacedby their tail behavior uniformly. The full power of this condition will be used in the nextchapter.

Proposition 3.1.1 Under conditions (C1)-(C4), the chain X is ϕ-irreducibility for allσ-finite measures ϕ(·) on B(R), such that ϕ λ.

Proof. We prove that for every x ∈ R and every B ∈ B(R), such that λ(B) > 0, thereexists n ∈ N, such that pn(x,B) > 0. It is enough to prove the claim in the case of boundedsets. Let B ∈ B(R), λ(B) > 0, be an arbitrary bounded set. Let x ∈ R and 0 < ε < 1 bearbitrary. Then, by (C2), there exists yε,x ≥ 1 such that for all |y| ≥ yε,x we have∣∣∣∣fx(y)

|y|α(x)+1

c(x)− 1

∣∣∣∣ < ε.

Furthermore, by (C3), there exists k > 0 such that for given ε there exists yε ≥ 1, suchthat for all |y| ≥ yε and all z ∈ [−k, k]c, we have∣∣∣∣fz(y)

|y|α(z)+1

c(z)− 1

∣∣∣∣ < ε.

3.1 General stable-like Markov chains 45

Let a := supB and y0 := (yε,x ∨ yε ∨ k) + |x|+ |a|+ 1. By (C4), we have

p2(x,B) =

∫Rp(x, dy)p(y,B) =

∫Rfx(y − x)

∫B−y

fy(z)dzdy

≥∫ 2y0

y0

fx(y − x)

∫B−y

fy(z)dzdy

>(1− ε)2c(x)

∫ 2y0

y0

(y − x)−α(x)−1c(y)

∫B−y|z|−α(y)−1dzdy

>(1− ε)2c(x)

(inf

y0≤y≤2y0c(y)

)∫ 2y0

y0

(y − x)−3

∫B−y|z|−3dzdy > 0,

since B − y ⊆ (−∞,−yε) for y ≥ y0. 2

Moreover, the maximal irreducibility measure is actually the Lebesgue measure.

Proposition 3.1.2 Conditions (C1)-(C4) imply that the maximal irreducibility measurefor the chain X is equivalent, in the absolutely continuous sense, with the Lebesgue mea-sure. Hence, the chain X is λ-irreducible.

Proof. Let ψ(·) be the maximal irreducibility measure of the chain X. By Proposition 3.1.1,the chain X is λ-irreducible, i.e., by the maximality of the measure ψ(·), λ ψ. Let usshow that ψ λ. If that would not be the case, i.e., if there would exist B ∈ B(R) suchthat λ(B) = 0 and ψ(B) > 0, then by irreducibility of the chain X, for every x ∈ R therewould exist n ∈ N such that

pn(x,B) =

∫Rp(x, dx1)

∫Rp(x1, dx2) . . .

∫Rp(xn−2, dxn−1)

∫B−xn−1

fxn−1(xn)dxn > 0.

But, since∫B−x fx(y)dy = 0, for every x ∈ R, because λ(B) = 0, we have pn(x,B) = 0. 2

Proposition 3.1.3 Condition (C3) implies

supx∈[−k,k]c

c(x) <∞.

Proof. Let 0 < ε < 1 be arbitrary. Then there exists yε ≥ 1 such that for all |y| ≥ yε wehave ∣∣∣∣fx(y)

|y|α(x)+1

c(x)− 1

∣∣∣∣ < ε for all x ∈ [−k, k]c, i.e.,

(1− ε) c(x)

|y|α(x)+1< fx(y) for all x ∈ [−k, k]c.

Therefore,

c(x) <1

1− ε

(2

∫ ∞yε

y−α(x)−1dy

)−1

≤ 1

1− ε

(2

∫ ∞yε

y−3dy

)−1

=y2ε

1− ε


for all x ∈ [−k, k]c. 2

Proposition 3.1.4 Conditions (C1)-(C4) imply that for the chain X every bounded Borelset S ⊆ [−k, k]c is a ν2-small set for some nontrivial measure ν2(·) on B(R).

Proof. By (C3), there exists k > 0, such that for all 0 < ε < 1 there exists yε ≥ k ∨ 1,such that for all |y| ≥ yε we have∣∣∣∣fx(y)

|y|α(x)+1

c(x)− 1

∣∣∣∣ < ε,

for all x ∈ [−k, k]c. Let S ⊆ (−∞,−k] be a bounded Borel set. Let x ∈ S and B ∈ B(R)be arbitrary. We have

p2(x,B) =

∫Rfx(y − x)

∫B−y

fy(z)dzdy ≥∫ 2yε

yε

fx(y − x)

∫(B−y)∩(−∞,−yε)

fy(z)dzdy

>(1− ε)2

(infx∈S

c(x)

)(inf

yε≤y≤2yεc(y)

)∫ 2yε

yε

(y − a)−3

∫(B−y)∩(−∞,−yε)

|z|−3dzdy,

where a := inf S. Now, by condition (C4), the measure

ν2(B) := (1− ε)2

(infx∈S

c(x)

)(inf

yε≤y≤2yεc(y)

)∫ 2yε

yε

(y − a)−3

∫(B−y)∩(−∞,−yε)

|z|−3dzdy

is a nontrivial measure. Therefore, the set S is a ν2-small set. 2

In the following proposition we prove that the chain X is aperiodic.

Proposition 3.1.5 Under conditions (C1)-(C4), the chain X is an aperiodic chain.

Proof. From the previous proposition we know that every bounded Borel set S ⊆ [−k, k]c

is a ν2-small set. Let us show that there exists a ν2-small set S ⊆ [−k, k]c which is also aν3-small set with ν3(·) = δ3ν2(·), for some δ3 > 0. Let S = [−4yε − k,−k], where ε > 0and yε ≥ k ∨ 1 are given as in the previous proposition. The set S is a ν2-small set. Let usshow that

infx∈S

p(x, S) > 0.

Then, by Theorem 2.7.16 (ii), S is a ν3-small set, where ν3(·) is a multiple of ν2(·). Wehave

p(x, S) =

∫S−x

fx(y)dy ≥∫

(S−x)∩(−∞,−yε)∪(S−x)∩(yε,∞)

fx(y)dy

>(1− ε)(

infx∈S

c(x)

)infx∈S

∫(S−x)∩(−∞,−yε)∪(S−x)∩(yε,∞)

|y|−3dy > 0.

2

As a direct consequence of the previous proposition and Theorem 2.7.19 we get the follow-ing.


Proposition 3.1.6 Conditions (C1)-(C4) imply that for the chain X, a Borel set is asmall set if, and only if, it is a petite set. 2

Unfortunately, we know that only compact subsets of [−k, k]c are petite sets. In order toprove petiteness of all compact sets, we introduce the following condition

(C5) there exists l > 0 such that for every compact set C ⊆ [−l, l]c with λ(C) > 0, wehave

infx∈[−k,k]

∫C−x

fx(y)dy > 0.

Proposition 3.1.7 Conditions (C1)-(C5) imply that for the chain X, every bounded Borelset is a petite set.

Proof. From Propositions 3.1.4, 3.1.5 and 3.1.6 we know that every bounded Borel setC ⊆ [−k, k]c is a petite set. Since the union of two petite sets is again a petite set (see [29,Proposition 5.5.5]), it is enough to show that [−k, k] is a petite set. Let C ⊆ (−∞,−k] bea bounded Borel set, i.e., a petite set. Let 0 < ε < 1 be arbitrary and let yε ≥ (k ∨ l ∨ 1)be such that for all |y| ≥ yε we have∣∣∣∣fx(y)

|y|α(x)+1

c(x)− 1

∣∣∣∣ < ε,

for all x ∈ [−k, k]c. Then, for every x ∈ [−k, k] we have

p2(x,C) =

∫Rfx(y − x)

∫C−y

fy(z)dzdy ≥∫ 2yε

yε

fx(y − x)

∫(C−y)∩(−∞,−yε)

fy(z)dzdy

> (1− ε)(

infyε≤y≤2yε

c(y)

)(∫(C−2yε−k)∩(−∞,−yε)

|z|−3dz

)inf

x∈[−k,k]

(∫[yε,2yε]−x

fx(y)dy

).

Now, using condition (C5), we have that infx∈[−k,k] p2(x,C) > 0. Therefore, by Theorem2.7.16 (ii) and Proposition 3.1.6, the set [−k, k] is a petite set, i.e., every bounded Borelset is a petite set. 2

As a consequence of the previous proposition, Theorem 2.7.8 (i) and Propositions 2.7.10 itfollows the following useful result.

Proposition 3.1.8 Conditions (C1)-(C5) imply that the chain X is

(i) a T-chain.

(ii) either H-recurrent or transient. 2

In the sequel, we call the chain X, given by the transition function p(x, dy) = fx(y−x)dywhich satisfies conditions (C1)-(C5), a stable-like Markov chain. Now, let us explainthe significance of each of conditions (C1)-(C5)


(C1) x 7−→ fx(y) is a Borel measurable function, for every y ∈ R

(C2) fx(y) ∼ c(x)|y|−α(x)−1, when |y| −→ ∞, for every x ∈ R


lim|y|−→∞

supx∈[−k,k]c

∣∣∣∣fx(y)|y|α(x)+1

c(x)− 1

∣∣∣∣ = 0

(C4) infx∈C

c(x) > 0 for every compact set C ⊆ [−k, k]c


infx∈[−k,k]

∫C−x

fx(y)dy > 0.

Clearly, condition (C1) ensures that the stable-like chain X is correctly defined whilecondition (C2) says that we deal only with chains with jump distributions similar to SαSdistributions. Condition (C3) is the strongest restriction here. It ensures that out of somecompact set, all jump densities can be replaced by their tail behavior at the same time.This condition is crucial in finding sufficient conditions for the recurrence and transienceof the stable-like chain X, i.e., it enables us to deal with the functions c(x)|y|−α(x)−1,instead of dealing with the density functions fx(y). Another chain property essential infinding sufficient conditions for the recurrence and transience is that every compact setis a petite set. This is the reason why compact sets are important in conditions (C3),(C4) and (C5). Condition (C4) ensures that the scaling function c(x) does not vanish onpetite sets (singletons), and condition (C5) ensures that the petite set (singleton) [−k, k]communicates with the rest of the state space.

As we mentioned, the recurrence and H-recurrence properties of Markov chains on thegeneral state space are not equivalent. The difference between these properties happens ona set of irreducibility measure zero (see Theorem 2.7.5). In the case of the stable-like chainX, it happens on a set of Lebesgue measure zero. Hence, if we change the stable-like chainX on a set of Lebesgue measure zero, it can happen that its recurrence and H-recurrenceproperties are not equivalent anymore. Let A ∈ B(R) be such that λ(A) = 0. Note that Acan be unbounded. Let X be a Markov chain on R given by the transition function

p(x, dy) = fx(y − x)dy,

where fx : x ∈ R is the family of density functions on R such that fx = fx for all x ∈ R\Aand it satisfies conditions (C1) and (C2). Note that conditions (C2)-(C5) are satisfied λ-a.e. Hence, it is easy to see that the chain X is λ-irreducible, the maximal irreducibilitymeasure is again the Lebesgue measure and the chain X is aperiodic. Therefore, a Borelset is a small set for X if, and only if, it is a petite set for X. But, we cannot concludethat every bounded Borel set is a petite set. The most we can get is that every boundedset B ∈ B(R \ A) is a petite set for X. As a consequence of this fact, we do not knowif the chain X is a T-chain, so we cannot deduce equivalence between recurrence and H-recurrence property of the chain X. But, since the chains X and X are λ-irreducible and


since they differ on the set with zero Lebesgue measure, the recurrence property of thestable-like chain X is equivalent with the recurrence property of the chain X, and the H-recurrence property of the stable-like chain X is equivalent with the H-recurrence propertyof the chain X. Hence, the chain X is recurrent if, and only if, it is H-recurrent. Indeed,let x ∈ R and B ∈ B(R) be arbitrary, then we have

U(x,B) = p(x,B) +

∫Rp(x, dy)U(y,B) = p(x,B) +

∫Rp(x, dy)U(y,B).

Hence, if X is recurrent, then X is also recurrent. The opposite claim is proved analogously.In the H-recurrence case, we have

L(x,B) = p(x,B) +

∫Bcp(x, dy)L(y,B) = p(x,B) +

∫Bcp(x, dy)L(y,B).

Therefore, the H-recurrence property of the stable-like chain X implies the H-recurrenceproperty of the chain X. The opposite claim is proved analogously.

In Proposition 3.1.7 it is proved that every bounded Borel set is a petite set for thestable-like chain X. Therefore, it is natural to expect that a change of the stable-like chainX on an arbitrary bounded Borel set will not affect its recurrence and transience property.Let B ∈ B(R) be bounded and let X be a stable-like Markov chain on R given by thetransition kernel

p(x, dy) = fx(y − x)dy,

where fx : x ∈ R is a family of density functions on R such that fx = fx for allx ∈ R \ B, and such that it satisfies conditions (C1)-(C5). Hence, the stable-like chain X

is λ-irreducible T-model, such that τXCd= τ XC for every starting point x ∈ Cc, where C is an

arbitrary compact set such that B ⊆ C and λ(C) > 0. Therefore, by Proposition 2.7.11,the stable-like chain X is H-recurrent if, and only if, the stable-like chain X is H-recurrent.

Remark 3.1.9 The stable-like chain X has asymptotically symmetric transition jumps.This assumption can be relaxed to the case of non-symmetric transition jumps. Thetransition densities of the stable-like chain X, from a state x, have a power-law decaywith exponent α(x) + 1. Let us now consider a Markov chain with transition densitieswith a power-law decay with exponent α−(x) + 1 on the left to the current state x andwith a power-law decay with exponent α+(x) + 1 on the right to the current state x. Letα+, α− : R −→ (0, 2) and c+, c− : R −→ (0,∞) be arbitrary functions and let X be aMarkov chain on R given by the transition kernel p(x, dy) = fx(y− x)dy, where fxx∈R isa family of density functions on R which satisfies

(C1) x 7−→ fx(y) is measurable, for every y ∈ R

(C2) fx(y) ∼ c+(x)y−α+(x)−1, when y −→∞, and

fx(y) ∼ c−(x)(−y)−α−(x)−1, when y −→ −∞



limy−→∞

supx∈[−k,k]c

∣∣∣∣fx(y)yα+(x)+1

c+(x)− 1

∣∣∣∣ = 0

and

limy−→−∞

supx∈[−k,k]c

∣∣∣∣fx(y)(−y)α−(x)+1

c−(x)− 1

∣∣∣∣ = 0

(C4) infx∈C

(c+(x) ∧ c−(x)) > 0 for every compact set C ⊆ [−k, k]c


infx∈[−k,k]

∫C−x

fx(y)dy > 0.

The chain X is also called a stable-like chain. It is clear that the stable-like chain X has thesame properties as the stable-like chain X, i.e., it is λ-irreducible, maximal irreducibilitymeasure is again the Lebesgue measure, it is aperiodic, every bounded Borel set is a petiteset for X and it is a T-chain.

3.2 Periodic stable-like Markov chains

Assume that the family of probability densities fxx∈R, from the definition of thestable-like chain X, satisfies the following four additional properties

(PC1) fx(−y) = fx(y) for all x, y ∈ R

(PC2) the function x 7−→ fx is a periodic function with period τ > 0

(PC3) the function (x, y) 7−→ fx(y) is continuous and strictly positive

(PC4) the functions α(x) and c(x) are B(R)/B(R) measurable.

Clearly, the periodicity of the function x 7−→ fx implies the periodicity of the functionsα(x) and c(x). Indeed, let x ∈ R be arbitrary, then we have

1 = lim|y|−→∞

fx+τ (y)|y|α(x+τ)+1

c(x+ τ)

= lim|y|−→∞

(fx(y)

|y|α(x)+1

c(x)

c(x)

c(x+ τ)|y|α(x+τ)−α(x)

)=

c(x)

c(x+ τ)lim|y|−→∞

|y|α(x+τ)−α(x).

Furthermore, (PC2) and (PC3) reduce conditions (C1)-(C5) to

3.3 (α, β)-stable-like Markov chains 51

(PC5) lim|y|−→∞

supx∈[0,τ ]

∣∣∣∣fx(y)|y|α(x)+1

c(x)− 1

∣∣∣∣ = 0

(PC6) infx∈[0,τ ]

c(x) > 0.

A Markov chain Xp given by the transition function p(x, dy) = fx(y − x)dy satisfyingconditions (PC1)-(PC6) is called a periodic stable-like Markov chain. Observe thatunder condition (PC5) Proposition 3.1.3 reads as follows

supx∈R

c(x) = supx∈[0,τ ]

c(x) <∞. (3.1)

Let us remark that by assuming B(R)/B(R) measurability of the function x 7−→ fx(y) forall y ∈ R, conditions (PC1), (PC3) and (PC4) are not essential for defining the periodicstable-like chain Xp. They will be needed in the proof of Theorem 4.3.1.

Remark 3.2.1 As in the general case, in the periodic case we can also consider the caseof non-symmetric transition jumps. Let α : R −→ (0, 2) and c−, c+ : R −→ (0,∞) be

arbitrary functions and let Xp

be a periodic stable-like Markov chain given by the transitionfunction p(x, dy) := fx(y−x)dy, where fxx∈R is a family of density functions on R whichsatisfies

(PC1) the function x 7−→ fx is a periodic function with period τ > 0

(PC2) the function (x, y) 7−→ fx(y) is continuous and strictly positive

(PC3) the functions α(x), c−(x) and c+(x) are B(R)/B(R) measurable

(PC4) limy−→−∞

supx∈[0,τ ]

∣∣∣∣fx(y)|y|α(x)+1

c−(x)− 1

∣∣∣∣ = 0 and limy−→∞

supx∈[0,τ ]

∣∣∣∣fx(y)yα(x)+1

c+(x)− 1

∣∣∣∣ = 0

(PC5) infx∈[0,τ ]

(c−(x) ∧ c+(x)) > 0.

3.3 (α, β)-stable-like Markov chains

The probability densities fxx∈R, in the definition of the stable-like chain X, are asymp-totically equivalent to stable probability densities. In the following proposition we treat thecase when the family of densities fxx∈R is exactly the family of Sα(x)(γ(x), δ(x)) densitiesand we give sufficient conditions on parameters α(x), γ(x) and δ(x), such that the familyof stable densities f(·;α(x), γ(x), δ(x))x∈R satisfies conditions (C1)-(C5).

Proposition 3.3.1 Let 0 < ε < 1, M > 0 and k ≥ 0 be arbitrary, and let Fα ⊆ [1, 2) andFγ ⊆ (0,∞) be arbitrary and finite. Furthermore, let

(i) α : R −→ (ε, 2 − ε) and α : R −→ (0, 1) ∪ Fα, such that infx∈C α(x) > 0 for allcompact sets C ⊆ R,


(ii) γ : R −→ (0,M), γ : R −→ Fγ and γ : R −→ (ε,M), such that infx∈C γ(x) > 0 forall compact sets C ⊆ R,

(iii) δ : R −→ (−M,M)

be arbitrary and B(R)/B(R) measurable. Define

α(x) :=

α(x), x ∈ [−k, k]α(x), x ∈ [−k, k]c,

and

γ(x) :=

γ(x), x ∈ [−k, k]

γ(x)1y:α(y)<1(x) + γ(x)1y:α(y)≥1(x), x ∈ [−k, k]c.

Then, for any l ≥ 0, the family of stable density functions f(·;α(x), γ(x), δ(x))x∈R sat-isfies conditions (C1)-(C5).

Proof. Condition (C1) is trivially satisfied since the functions α(x), γ(x) and δ(x) areB(R)/B(R) measurable, and

f(y;α(x), γ(x), δ(x)) =1

2π

∫Re−iyξeiδ(x)ξ−γ(x)|ξ|α(x)dξ

holds true for all x ∈ R. Condition (C2) is just the property of Sα(γ, δ) densities and it isgiven in Theorem 2.2.4 (i), (iii), (iv) and (v). Conditions (C3) and (C4) are also a simpleconsequences of Theorem 2.2.4 (i), (iii), (iv) and (v). Let us assume that the condition(C5) does not hold. Then, there exist l ≥ 0 and a compact set C ⊆ [−l, l]c with λ(C) > 0,such that

infx∈[−k,k]

∫C−x

f(y;α(x), γ(x), δ(x))dy = 0.

Hence, there exists a sequence xnn∈N ⊆ [−k, k], such that

limn−→∞

∫C

f(y − xn;α(xn), γ(xn), δ(xn))dy = 0.

Since α(x) = α(x) and γ(x) = γ(x) on [−k, k], by boundnes of the sequence xnn∈Nand the functions α(x), γ(x) and δ(x), there is a subsequence of the sequence xnn∈N,which we denote again by xnn∈N, such that xn −→ x0, α(xn) −→ α0, γ(xn) −→ γ0 andδ(xn) −→ δ0 as n −→∞, where x0 ∈ [−k, k], α0 ∈ [ε, 2−ε], γ0 ∈ [ε,M ] and δ0 ∈ [−M,M ].Hence, by the dominated convergence theorem we have

0 = limn−→∞

∫C

f(y − xn;α(xn), γ(xn), δ(xn))dy

=1

2πlimn−→∞

∫C

∫Re−i(y−xn)ξeiδ(xn)ξ−γ(xn)|ξ|α(xn)

dξdy

=1

2π

∫C

∫Re−i(y−x0)ξeiδ0ξ−γ0|ξ|

α0dξ =

∫C−x0

f(y;α0, γ0, δ0)dy.

This is impossible since λ(C) > 0. 2

3.3 (α, β)-stable-like Markov chains 53

Unfortunately, Proposition 3.3.1 does not cover the case when the stability functionα(x) takes infinitely many values in the interval [1, 2) since we do not know the seriesrepresentation of f(·;α, β, γ, δ) for α ≥ 1, as for α < 1 (see Theorem 2.2.4 (iv) and (v)).

Let α, β ∈ (0, 2) and γ, δ ∈ (0,∞) be arbitrary and let X(α,β) be a stable-like Markovchain determined by transition densities with the following characteristic exponents

ψ(x; ξ) =

γ|ξ|α, x < 0δ|ξ|β, x ≥ 0.

The stable-like chain X(α,β) jumps by a SαS distribution when it is on the left to the origin,and it jumps by a SβS distribution when it is on the right to the origin. The stable-likechain X(α,β) is called an (α, β)-stable-like Markov chain. Although, at the first glance,the (α, β)-stable-like chain X(α,β) seems to be very regular, such a Markov chain will not beeasy to deal with. For example, its corresponding semigroup does not satisfy the Cb-Fellerproperty and strong Feller property (lim infy−→0 p(y,O) ≥ p(0, O) does not hold for someopen sets O ⊆ R). Therefore, we introduce its “continuous” and in the recurrence and

transience sense equivalent version. Let k > 0 be arbitrary and let X(α,β)

be a Markovchain determined by transition densities with the following characteristic exponents

ψ(x; ξ) = γ(x)|ξ|α(x),

where functions α : R −→ (0, 2) and γ : R −→ (0,∞) are continuous functions such that

α(x) =

α, x < −kβ, x > k

and γ(x) =

γ, x < −kδ, x > k.

In other words, we have changed the stability and scaling functions of the (α, β)-stable-

like chain X(α,β) in a continuous way. Clearly, the chain X(α,β)

satisfies assumptions fromProposition 3.3.1, i.e., it is a stable-like chain, and, by Proposition 2.7.11, it is recurrentif, and only if, the (α, β)-stable-like chain X(α,β) is recurrent.

In general, let α : R −→ (0, 2) and γ : R −→ (0,∞) be B(R)/B(R) measurable and letXα(x) be a Markov chain determined by transition densities with following characteristicexponents

ψ(x; ξ) = γ(x)|ξ|α(x).

The chain Xα(x) does not satisfy the assumptions of Proposition 3.3.1, i.e., it satisfiesonly conditions (C1) and (C2). But, as we mentioned in Section 3.1, the chain Xα(x) isλ-irreducible. If, in addition, α(x) and γ(x) are continuous then we only have a problemwith condition (C3).

Proposition 3.3.2 If the functions α(x) and γ(x) are continuous, then the chain Xα(x)

is a T-chain. Hence, it is H-recurrent or transient. Moreover, its corresponding semigroupsatisfies the Cb-Feller property and strong Feller property.

Proof. Let us define a(·) := δ1(·) and T (x,B) := p(x,B) for x ∈ R and B ∈ B(R). Weprove that the function x 7−→ T (x,B) is lower semicontinuous for every B ∈ B(R). Letx ∈ R and B ∈ B(R) be arbitrary and such that λ(B) <∞. By the dominated convergence


theorem and the continuity of the functions α(x) and γ(x) we have

limy−→x

p(y,B) = limy−→x

∫B

f(z − y;α(y), γ(y))dz

= (2π)−1 limy−→x

∫B

∫R

cos(ξ(z − y))e−γ(y)|ξ|α(y)dξdz

= (2π)−1

∫B

∫R

cos(ξ(z − x))e−γ(x)|ξ|α(x)dξdz

= p(x,B).

Let B ∈ B(R) be arbitrary, then, by Fatou’s lemma, we have

lim infy−→x

p(y,B) = lim infy−→x

∑k∈Z

p(y,B ∩ (k, k + 1]) ≥∑k∈Z

p(x,B ∩ (k, k + 1]) = p(x,B). (3.2)

Hence, the chain Xα(x) is a T-chain. Equivalence between recurrence and H-recurrenceand the Cb-Feller property and strong Feller property of its corresponding semigroup fol-low from Proposition 2.7.10, (3.2) and Theorem 2.7.8 (iii) and (iv). 2

Chapter 4

Recurrence and transience ofstable-like Markov chains

In this chapter, we derive recurrence and transience conditions for stable-like Markovchains X, Xp and X(α,β), introduced in Chapter 3.

4.1 Recurrence of general stable-like Markov chains

In this section, we establish a recurrence condition for the stable-like chain X.

Theorem 4.1.1 Let α : R −→ (0, 2) be an arbitrary function such that

α := lim inf|x|−→∞

α(x) > 1.

Furthermore, let c : R −→ (0,∞) be an arbitrary function and let fxx∈R be a family ofdensity functions on R which satisfies conditions (C1)-(C5) and such that

lim supδ−→0

lim sup|x|−→∞

(1 + |x|)α(x)

c(x)

∫ δ(1+|x|)

−δ(1+|x|)ln

(1 + sgn (x )

y

1 + |x |

)fx(y)dy < R(α) (4.1)

when α < 2, where

R(α) :=∞∑i=1

1

i(2i− α)− ln 2

α− 1

2α

(Ψ

(α + 1

2

)−Ψ

(α2

))+γ

α+

Ψ(α)

α,

and the left-hand side in (4.1) is finite when α = 2. Then, the stable-like chain X isrecurrent. 2

The proof of Theorem 4.1.1 is completely based on the Foster-Lyapunov drift condition(Theorem 2.7.22) which says that in order to prove the recurrence property of the stable-like chain X, one has to find an appropriate petite set C and an appropriate unboundedof petite sets function V (x), which satisfies ∆V (x) =

∫R p(x, dy)V (y) − V (x) ≤ 0 for all

55

56 4. Recurrence and transience of stable-like Markov chains

x ∈ Cc. Beside finding appropriate test function V (x) and appropriate petite set C, theproblem is to prove the above mentioned function inequality on Cc. The idea is to findtest function V (x), such that the associated level sets CV (r) = y : V (y) ≤ r are compactsets, i.e., petite sets, and that CV (r) ↑ R when r −→ ∞. For the test function we chooseV (x) = ln(1 + |x|). Now, by proving that

lim sup|x|−→∞

|x|α(x)

c(x)

(∫Rp(x, dy)V (y)− V (x)

)< 0,

since compact sets are petite sets, the proof is accomplished.Before the proof of Theorem 4.1.1, let us discuss the condition in (4.1). The constant

R(α), appearing in (4.1), is strictly positive (see the proof of Theorem 4.1.1). Furthermore,Lemma 2.1.1 implies that R(α), as a function of α ∈ (1, 2), is strictly increasing and itsatisfies R(1) = 0 and limα−→2R(α) = ∞. The condition in (4.1) is needed to control thebehavior of the family of density functions fxx∈R on sets symmetric around the origin.Using ln(1 + x) ≤ x, (4.1) follows from the condition

lim supδ−→0

lim sup|x|−→∞

sgn (x )(1 + |x |)α(x)−1

c(x )

∫ δ(1+|x |)

−δ(1+|x |)y fx (y)dy < R(α). (4.2)

Furthermore, by (C3) and since α(x) ∈ (1, 2) for all |x| large enough, the relation in (4.2)is equivalent with

lim sup|x|−→∞

sgn (x )(1 + |x |)α(x)−1

c(x )

∫R

y fx (y)dy < R(α),

i.e., with

lim sup|x|−→∞

sgn (x )|x |α(x)−1

c(x )Ex [X1 − x ] < R(α). (4.3)

Indeed, let δ > 0 and 0 < ε < 1 be arbitrary. Then, by (C3), there exists yε > 0 such thatfor all |y| ≥ yε ∣∣∣∣fx(y)

|y|α(x)+1

c(x)− 1

∣∣∣∣ < ε

for all x ∈ [−k, k]c. By taking |x| ≥ yεδ− 1, we have (recall that α(x) ∈ (1, 2) for all |x|

large enough)∫Ry fx(y)dy =

∫ −δ(1+|x|)

−∞y fx(y)dy +

∫ δ(1+|x|)

−δ(1+|x|)y fx(y)dy +

∫ ∞δ(1+|x|)

y fx(y)dy

4.1 Recurrence of general stable-like Markov chains 57

>− (1 + ε)

∫ −δ(1+|x|)

−∞

c(x)

|y|α(x)dy +

∫ δ(1+|x|)

−δ(1+|x|)y fx(y)dy + (1− ε)

∫ ∞δ(1+|x|)

c(x)

|y|α(x)dy

=

∫ δ(1+|x|)

−δ(1+|x|)y fx(y)dy − 2εc(x)

(α(x)− 1)δα(x)−1(1 + |x|)α(x)−1.

In the same way we get∫Ry fx(y)dy <

∫ δ(1+|x|)


2εc(x)

(α(x)− 1)δα(x)−1(1 + |x|)α(x)−1.

Hence, we have

(1 + |x|)α(x)−1

c(x)

∫ δ(1+|x|)

−δ(1+|x|)y fx(y)dy − 2ε

(α(x)− 1)δα(x)−1<

(1 + |x|)α(x)−1

c(x)

∫Ry fx(y)dy

<(1 + |x|)α(x)−1

c(x)

∫ δ(1+|x|)


2ε

(α(x)− 1)δα(x)−1.

By taking lim sup|x|−→∞, lim supε−→0 and lim supδ−→0, respectively, we get the desiredresult, i.e.,

lim supδ−→0

lim sup|x|−→∞

sgn (x )(1 + |x |)α(x)−1

c(x )

∫ δ(1+|x |)

−δ(1+|x |)y fx (y)dy

= lim sup|x|−→∞

sgn (x )(1 + |x |)α(x)−1

c(x )

∫R

y fx (y)dy .

The relation in (4.3) actually says that when the stable-like chain X has moved far awayfrom the origin, since R(α) > 0, it cannot have strong tendency to move further from theorigin. Since R(α) > 0, it is clear that (4.3) is satisfied if α(x) ∈ (1, 2) for all |x| largeenough and if fx(y) = fx(−y) holds for all y ∈ R and for all |x| large enough. For anonsymmetric example one can take fx(y) to be the density function of an Sα−(γ−, 0, δ−)distribution, when x < 0, and the density function of an Sα+(γ+, 0, δ+) distribution, whenx ≥ 0, where α−, α+ ∈ (1, 2), γ−, γ+ ∈ (0,∞), δ− ≥ 0 and δ+ ≤ 0. In the proof of Theorem4.1.1 we need the following technical lemma.

Lemma 4.1.2 Let α : R −→ (1, 2) be an arbitrary function. Then for every R ≥ 0 wehave

lim|x|−→∞

1

2− α(x)

(1−

(|x||x|+R

)2−α(x))

= 0.

Proof. Let 0 < ε < 1 be arbitrary. Since

1

x(1− (1− ε)x) ≤ − ln(1− ε)


for all x ∈ (0, 1], we have

0 ≤ lim sup|x|−→∞

1

2− α(x)

(1−

(|x||x|+R

)2−α(x))

≤ lim sup|x|−→∞

1

2− α(x)

(1− (1− ε)2−α(x)

)≤ − ln(1− ε).

By letting ε −→ 0, we have the claim. 2

Proof of Theorem 4.1.1. The proof is divided in four steps.Step 1. In the first step, we explain our strategy of the proof. Let us define a function

V : R −→ R+ by the formulaV (x) := ln(1 + |x|).

From Proposition 3.1.7, the set CV (r) = y : V (y) ≤ r is a petite set for all r < ∞. Wewill show that there exists r0 > 0, big enough, such that

∫R p(x, dy)V (y) − V (x) ≤ 0 for

all x ∈ CcV (r0). Then, the desired result will follow from Theorem 2.7.22. Since CV (r) ↑ R,

when r −→∞, it is enough to show that

lim sup|x|−→∞

(1 + |x|)α(x)

c(x)

(∫Rp(x, dy)V (y)− V (x)

)< 0.

We have∫Rp(x, dy)V (y) =

∫Rfx(y − x)V (y)dy =

∫Rfx(y)V (y + x)dy

=

∫y+x>0

ln(1 + x+ y)fx(y)dy +

∫y+x<0

ln(1− x− y)fx(y)dy. (4.4)

Step 2. In the second step, we find an appropriate upper bound for the first summandin (4.4). For any x > 0 we have∫

y+x>0ln(1 + x+ y)fx(y)dy

= ln(1 + x)

∫y+x>0

fx(y)dy +

∫y+x>0

ln

(1 +

y

1 + x

)fx(y)dy.

Let 0 < δ < 1 be arbitrary. By restricting ln(1 + t) to intervals (−1,−δ), [−δ, δ], (δ, 1) and[1,∞), and using the Taylor expansion of the function ln(1 + t), i.e.,

ln(1 + t) =∞∑i=1

(−1)i+1

iti,


for t ∈ (−1, 1], we get∫y+x>0

ln(1 + x+ y)fx(y)dy ≤ ln(1 + x)

∫y+x>0

fx(y)dy

−∞∑i=1

1

i(1 + x)i

∫−1−x<y<−δ(1+x)∩y+x>0

|y|ifx(y)dy

+

∫−δ(1+x)≤y≤δ(1+x)∩y+x>0

ln

(1 +

y

1 + x

)fx(y)dy

+∞∑i=1

(−1)i+1

i(1 + x)i

∫δ(1+x)<y<1+x∩y+x>0

yifx(y)dy

+

∫y≥1+x∩y+x>0

ln

(1 +

y

1 + x

)fx(y)dy.

Furthermore, by taking x > δ1−δ we get∫

y+x>0ln(1 + x+ y)fx(y)dy ≤ ln(1 + x)

∫y+x>0

fx(y)dy

−∞∑i=1

1

i(1 + x)i

∫−x<y<−δ(1+x)

|y|ifx(y)dy

+

∫−δ(1+x)≤y≤δ(1+x)

ln

(1 +

y

1 + x

)fx(y)dy

+∞∑i=1

(−1)i+1

i(1 + x)i

∫δ(1+x)<y<1+x

yifx(y)dy

+

∫y≥1+x

ln

(1 +

y

1 + x

)fx(y)dy.

Let us put

U δ1 (x) :=− 1

1 + x

∫δ(1+x)<y<x

yfx(−y)dy +1

1 + x

∫δ(1+x)<y<1+x

yfx(y)dy

U δ2 (x) :=− 1

2(1 + x)2

∫δ(1+x)<y<x

y2fx(−y)dy − 1

2(1 + x)2

∫δ(1+x)<y<1+x

y2fx(y)dy

U δ3 (x) :=−

∞∑i=3

1

i(1 + x)i

∫δ(1+x)<y<x

yifx(−y)dy +∞∑i=3

(−1)i+1

i(1 + x)i

∫δ(1+x)<y<1+x

yifx(y)dy,

U δ4 (x) :=

∫−δ(1+x)≤y≤δ(1+x)

ln

(1 +

y

1 + x

)fx(y)dy, and

U5(x) :=

∫y≥1+x

ln

(1 +

y

1 + x

)fx(y)dy,


for 0 < δ < 1 and x > δ1−δ . Hence, we find∫ ∞

−xln(1 +x+y)fx(y)dy ≤ ln(1 +x)

∫ ∞−x

fx(y)dy+U δ1 (x) +U δ

2 (x) +U δ3 (x) +U δ

4 (x) +U5(x).

(4.5)Here comes the crucial step where condition (C3) is needed. In the above terms, by (C3),we can replace all the density functions fx(y) with the functions c(x)|y|−α(x)−1 and find amore operable upper bound in (4.5). Let 0 < ε < 1 be arbitrary. Then, by (C3), thereexists yε ≥ 1, such that for all |y| ≥ yε∣∣∣∣fx(y)

|y|α(x)+1

c(x)− 1

∣∣∣∣ < ε,

for all x ∈ [−k, k]c. Let x >(k ∨ yε−δ

δ∨ δ

1−δ

). By a straightforward computation, we have

U δ1 (x) <− (1− ε)c(x)

(α(x)− 1)(1 + x)α(x)

(δ−α(x)+1 −

(x

1 + x

)−α(x)+1)

+(1 + ε)c(x)

(α(x)− 1)(1 + x)α(x)

δ − δα(x)

δα(x)=: U δ,ε

1 (x),

U δ2 (x) <− (1− ε)c(x)

(1 + x)α(x)

1

2(2− α(x))

((x

1 + x

)2−α(x)

− δ2−α(x)

)

− (1− ε)c(x)

(1 + x)α(x)

1

2(2− α(x))

δα(x) − δ2

δα(x)=: U δ,ε

2 (x)

U δ3 (x) <− (1− ε)c(x)

(1 + x)α(x)

∞∑i=3

1

i(i− α(x))

((x

1 + x

)i−α(x)

− δi−α(x)

)

+c(x)

(1 + x)α(x)

∞∑i=3

((−1)i+1(1 + (−1)i+1ε)

i(i− α(x))

δα(x) − δi

δα(x)

)=: U δ,ε

3 (x) and

U5(x) <(1 + ε)c(x)

∫ ∞1+x

ln

(1 +

y

1 + x

)dy

yα(x)+1=: U ε

5 (x).

Hence, from (4.5), we get∫ ∞−x

ln(1+x+y)fx(y)dy < ln(1+x)

∫ ∞−x

fx(y)dy+U δ,ε1 (x)+U δ,ε

2 (x)+U δ,ε3 (x)+U δ

4 (x)+U ε5 (x).

(4.6)


Step 3. In the third step, we find an appropriate upper bound for the second summandin (4.4). We have∫

y+x<0ln(1− x− y)fx(y)dy

= ln(x− 1)

∫y+x<0

fx(y)dy +

∫y+x<0

ln

(−1− y

x− 1

)fx(y)dy.

Let x >(k ∨ yε−δ

δ∨ δ

1−δ

). Then, again by (C3),∫

y+x<0ln(1− x− y)fx(y)dy < ln(x− 1)

∫y+x<0

fx(y)dy

+ c(x)(1− ε)∫ 2x−2

x

ln

(−1 +

y

x− 1

)dy

|y|α(x)+1

+ c(x)(1 + ε)

∫ ∞2x−2

ln

(−1 +

y

x− 1

)dy

|y|α(x)+1

= ln(x− 1)

∫y+x<0

fx(y)dy

+ c(x)(1− ε)∫ ∞x

ln

(−1 +

y

x− 1

)dy

|y|α(x)+1

+ 2εc(x)

∫ ∞2x−2

ln

(−1 +

y

x− 1

)dy

|y|α(x)+1.

Let us put

U ε6 (x) :=c(x)(1− ε)

∫ ∞x

ln

(−1 +

y

x− 1

)dy

|y|α(x)+1

+ 2εc(x)

∫ ∞2x−2

ln

(−1 +

y

x− 1

)dy

|y|α(x)+1.

We have ∫y+x<0

ln(1− x− y)fx(y)dy < ln(x− 1)

∫y+x<0

fx(y)dy + U ε6 (x). (4.7)

Step 4. In the fourth step, we prove

lim supx−→∞

(1 + x)α(x)

c(x)

(∫Rp(x, dy)V (y)− V (x)

)< 0.

By combining (4.4), (4.6) and (4.7) we have∫Rp(x, dy)V (y) < U0(x) + U δ,ε

1 (x) + U δ,ε2 (x) + U δ,ε

3 (x) + U δ4 (x) + U ε

5 (x) + U ε6 (x),


where

U0(x) = ln(1 + x)

∫y+x>0

fx(y)dy + ln(x− 1)

∫y+x<0

fx(y)dy

= ln(1 + x)− ln(1 + x)

∫y+x<0

fx(y)dy + ln(x− 1)

∫y+x<0

fx(y)dy

< ln(1 + x) = V (x).

Hence, we have∫Rp(x, dy)V (y)− V (x) < U δ,ε

1 (x) + U δ,ε2 (x) + U δ,ε

3 (x) + U δ4 (x) + U ε

5 (x) + U ε6 (x). (4.8)

In the rest of the fourth step we prove

lim supx−→∞

(1 + x)α(x)

c(x)

(∫Rp(x, dy)V (y)− V (x)

)< lim sup

δ−→0lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U δ,ε

1 (x) + lim supδ−→0

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U δ,ε

2 (x)

+ lim supδ−→0

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U δ,ε

3 (x) + lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U ε

5 (x)

+ lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U ε

6 (x) +R(α) ≤ 0.

Recall that α = lim infx−→∞ α(x) > 1,

R(α) =∞∑i=1

1

i(2i− α)− ln 2

α− 1

2α

(Ψ

(α + 1

2

)−Ψ

(α2

))+γ

α+

Ψ(α)

α

and

lim supδ−→0

lim supx−→∞

(1 + x)α(x)

c(x)U δ

4 (x) < R(α)

when α < 2, and the above limit is finite when α = 2 (assumption (4.1)). We have

lim supδ−→0

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U δ,ε

1 (x)

= lim supδ−→0

lim supε−→0

lim supx−→∞

[− 1− εα(x)− 1

(δ−α(x)+1 −

(x

1 + x

)−α(x)+1)

+1 + ε

α(x)− 1

δ − δα(x)

δα(x)

]


= lim supδ−→0

lim supε−→0

lim supx−→∞

[− 1− εα(x)− 1

(δ − δα(x)

δα(x)+ 1−

(x

1 + x

)−α(x)+1)

+1 + ε

α(x)− 1

δ − δα(x)

δα(x)

]

= lim supδ−→0

lim supε−→0

lim supx−→∞

[2ε

α(x)− 1

δ − δα(x)

δα(x)− 1− εα(x)− 1

(1−

(x

1 + x

)−α(x)+1)]

= lim supx−→∞

[1

α(x)− 1

((x

x+ 1

)−α(x)+1

− 1

)]= 0. (4.9)

In the last two equalities we use the assumption lim inf |x|−→∞ α(x) > 1. From Lemma 4.1.2we have

lim supδ−→0

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U δ,ε

2 (x)

= lim supδ−→0

lim supε−→0

lim supx−→∞

[− 1− ε

2(2− α(x))

((x

1 + x

)2−α(x)

− δ2−α(x)

)

− 1− ε2(2− α(x))

δα(x) − δ2

δα(x)

]

= lim supδ−→0

lim supx−→∞

[− 1

2(2− α(x))

((x

1 + x

)2−α(x)

+δα(x) − δ2

δα(x)− 1

)

− 1

2(2− α(x))

δα(x) − δ2

δα(x)

]

= lim supδ−→0

lim supx−→∞

[− 1

2− α(x)

δα(x) − δ2

δα(x)

]≤− 1

2−α , α < 2

−∞, α = 2.(4.10)

Using the dominated convergence theorem, we have

lim supδ−→0

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U δ,ε

3 (x)

= lim supδ−→0

lim supε−→0

lim supx−→∞

[− (1− ε)

∞∑i=3

1

i(i− α(x))

((x

1 + x

)i−α(x)

− δi−α(x)

)

+∞∑i=3

(ε+ (−1)i+1)

i(i− α(x))

δα(x) − δi

δα(x)

]


= lim supδ−→0

lim supε−→0

lim supx−→∞

[∞∑i=3

−(

x1+x

)i−α(x)+ δi−α(x) + (−1)i+1 − (−1)i+1δi−α(x)

i(i− α(x))

+ ε

∞∑i=3

(x

1+x

)i−α(x) − δi−α(x) + 1− δi−α(x)

i(i− α(x))

]

= lim supδ−→0

lim supx−→∞

∞∑i=3

−(

x1+x

)i−α(x)+ δi−α(x) + (−1)i+1 − (−1)i+1δi−α(x)

i(i− α(x))

= lim supδ−→0

lim supx−→∞

∞∑i=3

(− ( x1+x

)i−α(x)+ (−1)i+1

i(i− α(x))+δi−α(x) − (−1)i+1δi−α(x)

i(i− α(x))

)≤ −

∞∑i=2

2

2i(2i− α)= −

∞∑i=2

1

i(2i− α). (4.11)

Therefore, by combining (4.9), (4.10) and (4.11) we get

lim supδ−→0

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)

(U δ,ε

1 (x) + U δ,ε2 (x) + U δ,ε

3 (x))

≤

−∞∑i=1

1

i(2i− α), α < 2

−∞, α = 2.

(4.12)

Now, let us compute

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U ε

5 (x).

Using integration by parts formula we get

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U ε

5 (x) = lim supx−→∞

(1 + x)α(x)

∫ ∞1+x

ln

(1 +

y

1 + x

)dy

yα(x)+1

= lim supx−→∞

(ln 2

α(x)+

1

α(x)

∫ ∞1

dy

yα(x)(1 + y)

).

Furthermore, from Lemma 2.1.1 and the fact that the function

x 7−→ Ψ

(x+ 1

2

)−Ψ

(x2

)is decreasing on (0,∞) (Lemma 2.1.1) we have

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U ε

5 (x)

= lim supx−→∞

(ln 2

α(x)+

1

2α(x)

(Ψ

(α(x) + 1

2

)−Ψ

(α(x)

2

)))


≤ ln 2

α+

1

2α

(Ψ

(α + 1

2

)−Ψ

(α2

)). (4.13)

At the end, using integration by parts formula, we have

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U ε

6 (x)

= lim supε−→0

lim supx−→∞

(1 + x)α(x)

[(1− ε)

∫ ∞x

ln

(−1 +

y

x− 1

)dy

|y|α(x)+1

+ 2ε

∫ ∞2x−2

ln

(−1 +

y

x− 1

)dy

|y|α(x)+1

]

= lim supε−→0

lim supx−→∞

(1 + x)α(x)

[1− εα(x)

(1

xα(x)ln

(−1 +

x

x− 1

)+

∫ ∞x

dy

yα(x)(y − x+ 1)

)

+2ε

α(x)

∫ ∞2x−2

dy

yα(x)(y − x+ 1)

]

= lim supε−→0

lim supx−→∞

[1− εα(x)

((1 + x)α(x)

xα(x)ln

(−1 +

x

x− 1

)+

(1 + x)α(x)

(x− 1)α(x)

∫ x−1x

0

yα(x)−1

1− ydy

)

+2ε

α2(x)

(1 + x)α(x)

(x− 1)α(x) 2F1(α(x), α(x), α(x) + 1;−1)

],

where in the last equality we use (2.3). From (2.3) we get

2F1(α(x), α(x), α(x) + 1;−1) ≤ 2

∫ 1

0

(1 + t)−1dt = ln 4.

Further,

limx−→∞

(ln

(−1 +

x

x− 1

)+

∫ x−1x

0

yα(x)−1

1− ydy + γ + Ψ(α(x))

)

= limx−→∞

(ln

(−1 +

x

x− 1

)+

∫ x−1x

0

yα(x)−1

1− ydy +

∫ 1

0

1− yα(x)−1

1− y

)

= limx−→∞

(ln

(−1 +

x

x− 1

)+

∫ x−1x

0

1

1− ydy +

∫ 1

x−1x

1− yα(x)−1

1− y

)= 0.


Hence,

lim supε−→0

lim supx−→∞

(1 + x)α(x)

c(x)U ε

6 (x)

≤ lim supx−→∞

1

α(x)

((1 + x)α(x)

xα(x)ln

(−1 +

x

x− 1

)+

(x+ 1)α(x)

(x− 1)α(x)

∫ x−1x

0

yα(x)−1

1− ydy

)

= −γα− Ψ(α)

α. (4.14)

By combining (4.8), (4.12), (4.13) and (4.14) we have

lim supx−→∞

(1 + x)α(x)

c(x)

(∫Rp(x, dy)V (y)− V (x)

)< 0.

The case when x < 0 is treated in the same way. Therefore, we have proved the desiredresult. 2

4.2 Transience of general stable-like Markov chains

In this section, we establish a transience condition for the stable-like chain X.

Theorem 4.2.1 Let α : R −→ (0, 2) be an arbitrary function, such that

α := lim sup|x|−→∞

α(x) < 1

and let β ∈ (0, 1 − α) be arbitrary. Furthermore, let c : R −→ (0,∞) be an arbitraryfunction and let fxx∈R be a family of density functions which satisfies conditions (C1)-(C5) and there exists a0 > 0, such that

lim inf|x|−→∞

α(x)|x|α(x)

c(x)

∫ a

−a

(1−

(1 + sgn (x )

y

1 + |x |

)−β)fx(y)dy > −T (α, β) (4.15)

for all a ≥ a0, where

T (α, β) := 2F1(−α, β, 1− α; 1) + βB(1;α + β, 1− α)− αB(1;α + β, 1− β).

Then, the stable-like chain X is transient. 2

Analogously as in the case of recurrence, the proof of Theorem 4.2.1 is again based on theFoster-Lyapunov drift condition (Theorem 2.7.20) which says that in order to prove thetransience property of the stable-like chain X, one has to find an appropriate bounded andB(R)/B(R+) measurable function V (x) and a constant r ≥ 0, such that CV (r), Cc

V (r) ∈B+(R) and ∆V (x) =

∫R p(x, dy)V (y) − V (x) ≥ 0 for all x ∈ Cc

V (r). Again, beside finding

4.2 Transience of general stable-like Markov chains 67

appropriate test function V (x) and appropriate constant r ≥ 0, the problem is to prove theabove mentioned function inequality on Cc

V (r). The idea is to find test function V (x) suchthat the associated level sets CV (r) are compact sets, i.e., petite sets, and that CV (r) ↑ R,when r −→ 1 (this is possible since V (x) is bounded). For the test function we takeV (x) = 1− (1 + |x|)−β. By proving that

lim inf|x|−→∞

α(x)|x|α(x)+β

c(x)

(∫Rp(x, dy)V (y)− V (x)

)> 0,

since compact sets are petite sets, the proof is accomplished.Before the proof of Theorem 4.2.1, let us discuss the condition in (4.15). Using properties

of hypergeometric functions and the Beta function, it is easy to see that the constantT (α, β), as a function of β ∈ (0, 1−α) for fixed α ∈ (0, 1), is strictly positive and T (α, 0) =T (α, 1 − α) = 0, while considered as a function of α ∈ [0, 1 − β) for fixed β ∈ (0, 1), itis strictly decreasing, T (0, β) = 2 and T (1 − β, β) = 0. As in the recurrent case, (4.15) isneeded to control the behavior of the family of density functions fxx∈R on sets symmetricaround the origin. Using the concavity property of the function x 7−→ xβ, for β ∈ (0, 1−α),condition (4.15) follows from the condition

lim sup|x|−→∞

α(x)

c(x)|x|α(x)−1 <

T (α, β)

a0β. (4.16)

Indeed, we have

lim inf|x|−→∞

α(x)|x|α(x)

c(x)

∫ a

−a

(1−

(1 + sgn (x )

y

1 + |x |

)−β)fx(y)dy

≥ lim inf|x|−→∞

α(x)|x|α(x)

c(x)

(1 + |x| − a)β − (1 + |x|)β

(1 + |x| − a)β

∫ a

−afx(y)dy

≥ lim inf|x|−→∞

(− aβα(x)|x|α(x)

c(x)(1 + |x| − a)

)= −aβ lim sup

|x|−→∞

α(x)

c(x)|x|α(x)−1,

where in the second inequality we used

(1 + |x| − a)β − (1 + |x|)β

a≥ − ∂

∂x(1 + |x| − a)β = −β(1 + |x| − a)β−1.

Note that the relation in (4.16) actually says that the function c(x) cannot decrease toofast. Since T (α, β) > 0 and α(x) ∈ (0, 1) for all |x| large enough, a simple example whichsatisfies (4.16) is the case when c(x) ≥ d|x|α(x)−1+ε, for some d > 0 and for all |x| largeenough, where 0 < ε < 1 − α is arbitrary. Furthermore, one can prove that the functionβ 7−→ T (α, β)/β is strictly decreasing on (0, 1− α). Hence, according to (4.16), we chooseβ close to 0.


Proof of Theorem 4.2.1. The proof is divided in three steps.Step 1. In the first step, we explain our strategy of the proof. Let us define a function

V : R −→ R+ by the formula

V (x) := 1− (1 + |x|)−β,

where 0 < β < 1 − α is as in the statement of the theorem (recall that α =lim supx−→∞ α(x) < 1). It is clear that CV (r) ∈ B+(R) and Cc

V (r) ∈ B+(R), for every0 < r < 1. By Theorem 2.7.20, we have to show that there exists 0 < r0 < 1 such that∆V (x) ≥ 0, for every x ∈ Cc

V (r0). Since CV (r) ↑ R, when r ↑ 1, it is enough to show that

lim inf|x|−→∞

α(x)|x|α(x)+β

c(x)

(∫Rp(x, dy)V (y)− V (x)

)> 0.

We have∫Rp(x, dy)V (y)− V (x) =

∫RV (y + x)fx(y)dy − V (x)

=

∫y+x>0

(1− (1 + y + x)−β

)fx(y)dy +

∫y+x<0

(1− (1− y − x)−β

)fx(y)dy

−(1− (1 + |x|)−β

) ∫y+x>0

fx(y)dy −(1− (1 + |x|)−β

) ∫y+x<0

fx(y)dy

= (1 + |x|)−β[∫ ∞−x

(1−

(1 + |x|

1 + x+ y

)β)fx(y)dy +

∫ −x−∞

(1−

(1 + |x|

1− x− y

)β)fx(y)dy

].

(4.17)

Step 2. In the second step, by using condition (C3), we find an operable lower boundfor (4.17). First, let us take a look at the case when x > 0. Let 0 < ε < 1 be arbitrary.Then, by (C3), there exists yε ≥ a0∨ 1 (the constant a0 > 0 is defined in (4.15)), such thatfor all |y| ≥ yε ∣∣∣∣fx(y)

|y|α(x)+1

c(x)− 1

∣∣∣∣ < ε,

for all x ∈ [−k, k]c. Let x ≥ k ∨ yε. Then we have∫ ∞−x

(1−

(1 +

y

1 + x

)−β)fx(y)dy > c(x)(1 + ε)

∫ x

yε

(1−

(1− y

1 + x

)−β)dy

yα(x)+1

+

∫ yε

−yε

(1−

(1 +

y

1 + x

)−β)fx(y)dy + c(x)(1− ε)

∫ ∞yε

(1−

(1 +

y

1 + x

)−β)dy

yα(x)+1

and∫ −x−∞

(1−

(1 + x

1− x− y

)β)fx(y)dy >

c(x)(1− ε)α(x)xα(x)

− c(x)(1 + ε)

∫ ∞x

(1 + x

1− x+ y

)βdy

yα(x)+1.


Note that this was the crucial step where we needed condition (C3). For given 0 < ε < 1and x ≥ k ∨ yε let us put

U ε1 (x) := c(x)(1 + ε)

∫ x

yε

(1−

(1− y

1 + x

)−β)dy

yα(x)+1,

U ε2 (x) :=

∫ yε

−yε

(1−

(1 +

y

1 + x

)−β)fx(y)dy,

U ε3 (x) := c(x)(1− ε)

∫ ∞yε

(1−

(1 +

y

1 + x

)−β)dy

yα(x)+1,

U ε4 (x) :=

c(x)(1− ε)α(x)xα(x)

and

U ε5 (x) := c(x)(1 + ε)

∫ ∞x

(1 + x

1− x+ y

)βdy

yα(x)+1.

Hence, we have∫Rp(x, dy)V (y)− V (x) > U ε

1 (x) + U ε2 (x) + U ε

3 (x) + U ε4 (x)− U ε

5 (x).

Step 3. In the third step, we prove

lim infx−→∞

α(x)xα(x)+β

c(x)

(∫Rp(x, dy)V (y)− V (x)

)> lim inf

ε−→0lim infyε−→∞

lim infx−→∞

α(x)xα(x)

c(x)(U ε

1 (x) + U ε3 (x) + U ε

4 (x)− U ε5 (x))− T (α, β) ≥ 0.

(4.18)

Recall that

T (α, β) = 2F1(−α, β, 1− α; 1) + βB(1;α + β, 1− α)− αB(1;α + β, 1− β)

and

lim infε−→0

lim infyε−→∞

lim infx−→∞

α(x)xα(x)

c(x)U ε

2 (x) > −T (α, β)

(assumption (4.15)). By straightforward computations, using (2.3), (2.6) and (2.8), wehave

α(x)xα(x)

c(x)U ε

1 (x) =(1 + ε)xα(x)

yα(x)ε

−(1 + ε)xα(x)

2F1

(−α(x), β, 1− α(x); yε

1+x

)yα(x)ε

− (1 + ε) + (1 + ε) 2F1

(−α(x), β, 1− α(x);

x

1 + x

),


α(x)xα(x)

c(x)U ε

4 (x) = (1− ε) and

α(x)xα(x)

c(x)U ε

5 (x) =(1 + ε)α(x)xα(x)(1 + x)βB

(x−1x

;α(x) + β, 1− β)

(x− 1)α(x)+β.

It is easy to check that

∂

∂y

(−2F1

(−α(x), β, 1− α(x);− y

1+x

)α(x)yα(x)(1 + x)β

)=

1

(1 + x+ y)βyα(x)+1,

and from (2.1) and (2.7) we have

2F1

(−α(x), β, 1− α(x);− y

1+x

)α(x)yα(x)(1 + x)β

=2F1

(β, α(x) + β, 1 + α(x) + β;−1+x

y

)(α(x) + β)yα(x)+β

+Γ(1− α(x))Γ(α(x) + β)

α(x)(1 + x)2α(x)+βΓ(β).

Therefore,

∫dy

(1 + x+ y)βyα(x)+1= −

2F1

(β, α(x) + β, 1 + α(x) + β;−1+x

y

)(α(x) + β)yα(x)+β

,

i.e.,

α(x)xα(x)

c(x)U ε

3 (x)

=(1− ε)xα(x)

yα(x)ε

−(1− ε)α(x)xα(x)(1 + x)β 2F1

(β, α(x) + β, 1 + α(x) + β,−1+x

yε

)yα(x)+βε (α(x) + β)

.

Furthermore, from (2.1), (2.4) and (2.7) we have

α(x)xα(x)

c(x)U ε

3 (x)

=(1− ε)xα(x)

yα(x)ε

− (1− ε)xα(x)

yα(x)ε

2F1

(β,−α(x), 1− α(x);− yε

x+ 1

)− (1− ε)Γ(α(x) + β)Γ(−α(x))α(x)xα(x)

Γ(β)(1 + x)α(x) 2F1

(α(x) + β, 0, 1 + α(x);− yε

x+ 1

)=

(1− ε)xα(x)

yα(x)ε

− (1− ε)xα(x)

yα(x)ε

2F1

(β,−α(x), 1− α(x);− yε

x+ 1

)− (1− ε)Γ(α(x) + β)Γ(−α(x))α(x)xα(x)

Γ(β)(1 + x)α(x).


Let us put

V ε1 (x) :=

(1 + ε)xα(x)

yα(x)ε

−(1 + ε)xα(x)

2F1

(−α(x), β, 1− α(x); yε

1+x

)yα(x)ε

V ε2 (x) :=

(1− ε)xα(x)

yα(x)ε

− (1− ε)xα(x)

yα(x)ε

2F1

(β,−α(x), 1− α(x);− yε

x+ 1

)and

V ε3 (x) :=(1 + ε) 2F1

(−α(x), β, 1− α(x);

x

1 + x

)− (1− ε)Γ(α(x) + β)Γ(−α(x))α(x)xα(x)

Γ(β)(1 + x)α(x)

−(1 + ε)α(x)xα(x)(1 + x)βB

(x−1x

;α(x) + β, 1− β)

(x− 1)α(x)+β.

Hence, (4.18) is reduced to

lim infx−→∞

α(x)xα(x)+β

c(x)

(∫Rp(x, dy)V (y)− V (x)

)> lim inf


lim infx−→∞

V ε1 (x) + lim inf


lim infx−→∞

V ε2 (x) + lim inf

ε−→0lim infx−→∞

V ε3 (x)− T (α, β).

(4.19)

By (2.2) and (2.3), we have

0 ≤ 2F1

(−α(x), β, 1− α(x),

yε1 + x

)≤ 1,

thereforelim infε−→0

lim infyε−→∞

lim infx−→∞

V ε1 (x) ≥ 0. (4.20)

Since 1 − α(x) − (−α(x)) − β = 1 − β > 0, from (2.2) and the dominated convergencetheorem, we have

lim infε−→0

lim infyε−→∞

lim infx−→∞

V ε2 (x) = 0. (4.21)

At the end, let us computelim infε−→0

lim infx−→∞

V ε3 (x).

From (2.1) we have

2F1

(−α(x), β, 1− α(x);

x

1 + x

)≥ 2F1 (−α(x), β, 1− α(x); 1) ,

and from (2.8) we have

α(x)B

(x− 1

x;α(x) + β, 1− β

)≤ α(x)B (1;α(x) + β, 1− β) .


Hence, we have

lim infε−→0

lim infx−→∞

V ε3 (x) ≥ lim inf

ε−→0lim infx−→∞

[(1 + ε) 2F1(−α(x), β, 1− α(x); 1)

− (1− ε)α(x)Γ(α(x) + β)Γ(−α(x))xα(x)

Γ(β)(1 + x)α(x)

− (1 + ε)α(x)B(1;α(x) + β, 1− β)xα(x)(1 + x)β

(1− x)α(x)+β

],

i.e., since all terms are bounded,

lim infε−→0

lim infx−→∞

V ε3 (x)

≥ lim infx−→∞

[2F1(−α(x), β, 1− α(x); 1) + βB(1;α(x) + β, 1− α(x))

− α(x)B(1;α(x) + β, 1− β)

].

One can prove that the function

y 7−→ T (y, β) :=2 F1(−y, β, 1− y; 1) + βB(1; y + β, 1− y)− yB(1; y + β, 1− β)

is strictly decreasing on [0, 1 − β), and it easy to see that T (1 − β, β) = 0. Hence, since0 ≤ α < 1− β, we have

lim infε−→0

lim infx−→∞

V ε3 (x) ≥ T (α, β). (4.22)

By combining (4.19), (4.20), (4.21) and (4.22) we have

lim infx−→∞

α(x)xα(x)+β

c(x)

(∫Rp(x, dy)V (y)− V (x)

)> 0.

The case when x < 0 is treated in the same way. Therefore, by Theorem 2.7.20, the stable-like chain X is transient. 2

4.3 Recurrence and transience of periodic stable-like

Markov chains

In this section, we derive a recurrence and transience criterion for the periodic stable-likechain Xp.

Theorem 4.3.1 If the set x : α(x) = α0 := infx∈R α(x) has positive Lebesgue measure,then the stable-like chain Xp is recurrent if, and only if, α0 ≥ 1. 2

4.3 Recurrence and transience of periodic stable-like Markov chains 73

The idea of the proof is to subordinate the stable-like chain Xp with a Poisson processN1 with parameter κ = 1. In this way we get a Markov semimartingale and, by applying

Theorem 2.6.3, we prove that the sequence of processes Ypn := n−

1α0Xp

N1ntt∈R+ , n ∈ N,

converges in distribution to a symmetric α0-stable Levy process. Furthermore, we provethat all the processes Yp

n, n ∈ N, are either recurrent or transient at the same time, theirrecurrence property is equivalent with the recurrence property of a symmetric α0-stableLevy process and the recurrence properties of the process Yp

1 and the stable-like chain Xp

are equivalent. This accomplishes the proof. Before the proof of Theorem 4.3.1, we provesome auxiliary results.

Proposition 4.3.2 Let a 6= 0 be arbitrary and let Nκ be a Poisson process with pa-rameter κ > 0 independent of an Rd-valued Markov chain Y. Then, the processY(a,κ) := aYNκ

tt∈R+ is

(i) a Markov semimartingale, its corresponding semigroup P (a,κ)t t∈R+ is strongly con-

tinuous and infinitesimal generator has the form

A(a,κ)f(x) = κ

∫R(f(y)− f(x))p

(a−1x, a−1dy

)with domain DA(a,κ) = Bb(Rd)

(ii) λ-irreducible and recurrent (respectively H-recurrent) if, and only if, the chain Y isλ-irreducible and recurrent (respectively H-recurrent).

(iii) In the case when the chain Y is given by transition function p(x, dy) := fx(y− x)dy,where fxx∈Rd is a family of probability densities on Rd such that x 7−→ fx(y) isB(Rd)/B(R) measurable function for all y ∈ Rd, the process Y(a,κ) is a homogeneousdiffusion with jumps whose characteristics and modified characteristics, relative to atruncation function h(x), are given by

Bi(a,κ)t = a−1κ

∫ t

0

∫Rdhi(y)fYNκs (a−1y)dyds,

Cij(a,κ)t = 0,

Cij(a,κ)t = a−1κ

∫ t

0

∫Rdhi(y)hj(y)fYNκs (a−1y)dyds and

N (a,κ)(ds, dy) = a−1κfYNκs (a−1y)dyds.

Proof.

(i) First, note that if Y is a Markov chain with respect to the family of probabilitymeasures Pxx∈Rd , then aY := aYnn∈Z+ is a Markov chain with respect to the

family of probability measures Qx := Pa−1xx∈Rd . Hence, the process Y(a,κ) is a


Markov semimartingale. Its transition function is given by

p(a,κ)t (x, dy) = e−κt

∞∑k=0

(κt)k

k!pk(a−1x, a−1dy

),

and semigroup operators are given by

P(a,κ)t f(x) =

∫Rdp

(a,κ)t

(a−1x, a−1dy

)f(y) = e−κt

∞∑k=0

(κt)k

k!

∫Rdpk(a−1x, a−1dy

)f(y),

for f ∈ Bb(Rd) and t ∈ R+.

Let us prove the strong continuity property of the semigroup P (a,κ)t t∈R+ . For f ∈

Bb(Rd) we have

|P (a,κ)t f(x)− f(x)| ≤ e−κt

∞∑k=0

(κt)k

k!

∫Rdpk(a

−1x, a−1dy)|f(y)− f(x)|

= e−κt∞∑k=1

(κt)k

k!

∫Rdpk(a

−1x, a−1dy)|f(y)− f(x)|

≤ 2Me−κt(eκt − 1),

where the constant M ≥ 0 is such that ||f ||∞ ≤M . Therefore,

limt−→0||P (a,κ)

t f − f ||∞ = 0.

Define an operator A(a,κ) : Bb(Rd) −→ Bb(Rd) by

A(a,κ)f(x) := κ

∫Rd

(f(y)− f(x))p(a−1x, a−1dy

).

For f ∈ Bb(Rd) and x ∈ Rd we have∣∣∣∣∣P (a,κ)t f(x)− f(x)

t−A(a,κ)f(x)

∣∣∣∣∣=

∣∣∣∣∣P (a,κ)t f(x)− f(x)

t− κ

∫Rd


)∣∣∣∣∣=∣∣∣e−κt ∞∑

k=1

κktk−1

k!

∫Rdpk(a

−1x, a−1dy)(f(y)− f(x))

− κ∫Rd


) ∣∣∣


≤ κ(e−λt − 1)

∫Rdp(a−1x, a−1dy)|f(y)− f(x)|

+ e−κt∞∑k=2

κktk−1

k!

∫Rdpk(a

−1x, a−1dy)|f(y)− f(x)|

≤ 2Mκ(e−κt − 1) + 2Me−κt∞∑k=2

κktk−1

k!,

where the constant M ≥ 0 is such that ||f ||∞ ≤M . Therefore,

limt−→0

∣∣∣∣∣∣∣∣∣∣P (a,κ)

t f − ft

−A(a,κ)f

∣∣∣∣∣∣∣∣∣∣∞

= 0,

i.e., the infinitesimal generator of the semigroup P (a,κ)t t∈R+ is

A(a,κ)f(x) = κ

∫Rd


),

with DA(a,κ) = Bb(Rd).

(ii) Let x ∈ Rd and B ∈ B(Rd) be arbitrary. We have

ExQ[∫ ∞

0

1Y (a,κ)t ∈Bdt

]=∞∑n=0

pn(a−1x, a−1B

) ∫ ∞0

e−κt(κt)n

n!dt

=1

κEa−1x

P

[∞∑n=0

1Yn∈a−1B

],

where in the second equality we used∫ ∞0

e−κt(κt)ndt =n!

κ.

Therefore, the process Y(a,κ) is λ-irreducible and recurrent if, and only if, the chainY is λ-irreducible and recurrent. In the H-recurrence case we have

L(a,κ)(x,B) = Qx(τ(a,κ)B <∞) = Pa−1x(τa−1B <∞) = L(a−1x, a−1B).

Hence, Y(a,κ) is H-recurrent if, and only if, the chain Y is H-recurrent.

(iii) By (i), the infinitesimal generator of the process Y(a,κ) is given by A(a,κ)f(x) =a−1κ

∫Rd(f(y + x) − f(x))fa−1x(a

−1y)dy, f ∈ Bb(Rd). Furthermore, by Theorem2.5.2, for every f ∈ Bb(Rd) the process

M ft := f(Y

(a,κ)t )− f(Y

(a,κ)0 )−

∫ t

0

A(a,κ)f(Y(a,κ)s− )ds


is a martingale. Let h(x) be a truncation function and let f ∈ C1b (Rd). Then,

M ft t∈R+ can be rewritten in the following form

M ft = f(Y

(a,κ)t )− f(Y

(a,κ)0 )− κ

a

∫ t

0

∫Rd

(f(y + Y

(a,κ)s− )− f(Y

(a,κ)s− )

)fYNκs−

(ya

)dyds

= f(Y(a,κ)t )− f(Y

(a,κ)0 )− κ

a

∫ t

0

∫Rd

d∑i=1

∂

∂xif(Y

(a,κ)s− )hi(y)fYNκs−

(ya

)dyds

− κ

a

∫ t

0

∫Rd

(f(y + Y

(a,κ)s− )− f(Y

(a,κ)s− )−

d∑i=1

∂

∂xif(Y

(a,κ)s− )hi(y)

)fYNκs−

(ya

)dyds.

Now, from Theorems 2.5.14 and 2.5.15, the claim follows. 2

Recall that the functions x 7−→ fx, α(x) and c(x) are τ -periodic, the function (x, y) 7−→fx(y) is continuous and strictly positive and α(x) and c(x) are B(R)/B(R) measurable.Let us put Λ := τZ and let ΠΛ : R −→ R/Λ be the covering map. We denote by XΛp

the process on R/Λ obtained by the projection of the stable-like chain Xp with respect toΠΛ(x).

Lemma 4.3.3 [25, Proposition 3.8.8] [3, Theorem III.3.1] The process XΛp is a universalMarkov chain on R/Λ determined by a transition density function of the form

pΛ(x, y) =∑k∈Λ

p(zx, zy + k) =∑k∈Λ

fzx(zy − zx + k)

for all x, y ∈ R/Λ, where zx and zy are arbitrary points in Π−1Λ (x) and Π−1

Λ (y), respec-tively. Furthermore, the chain XΛp possesses an invariant measure π(·), with π(R/Λ) <∞,and there exist constants C > 0 and c > 0, such that for all τ -periodic functions f ∈ Bb(R)we have∫

R/Λf(zx)π(dx) = 0 =⇒

∣∣∣∣∣∣∣∣∫Rpk(·, dy)f(y)

∣∣∣∣∣∣∣∣∞≤ C||f ||∞e−ck for all k ∈ N.

2

Since π(R/Λ) < ∞, without loss of generality, we assume that π(R/Λ) = 1. Followingthe ideas from the proof of [15, Theorem 1], we give a proof of Theorem 4.3.1.

Proof of Theorem 4.3.1. Let XΛp be as in Lemma 4.3.3. Let us suppose thatthe set x ∈ R : α(x) = α0 := infx∈R α(x) has positive Lebesgue measure. By λ-irreducibility of the stable-like chain Xp, this is equivalent with π(ΠΛ(x ∈ R : α(x) =α0 := infx∈R α(x))) > 0. Indeed, since π(·) is an invariant measure of the chain XΛp,∫

R/ΛpΛ(x,B)π(dx) = π(B)


holds for all B ∈ B(R/Λ), where B(R/Λ) denotes the Borel σ-algebra with respect to thequotient topology. Let us put A := x ∈ R : α(x) = α0 := infx∈R α(x) and B := ΠΛ(A).We have

π(B) =

∫R/Λ

pΛ(x,B)π(dx) =

∫R/Λ

p(zx,Π−1Λ (B))π(dx) =

∫R/Λ

p(zx, A)π(dx).

Now, if λ(A) > 0, then p(zx, A) > 0 for all zx ∈ R. Therefore, π(B) > 0 as well. On theother hand, if λ(A) = 0, then p(zx, A) = 0 for all zx ∈ R. Hence, π(B) = 0.

In the sequel (because of τ -periodicity) we use the abbreviation α(x) and c(x), for α (zx)and c (zx), where x ∈ R/Λ and zx ∈ Π−1

Λ (x) are arbitrary.Let N1 be a Poisson process with parameter κ = 1 independent of the stable-like

chain Xp and let us define a λ-irreducible Markov semimartingale Yp := XpNtt∈R+ . By

Proposition 4.3.2 (i), the semigroup of the process Yp is given by

Ptf(x) = e−t∞∑k=0

tk

k!

∫Rpk(x, dy)f(y)

for f ∈ Bb(R) and t ∈ R+. Hence, by Lemma 4.3.3, for every τ -periodic function f ∈ Bb(R)we have

||Ptf ||∞ ≤ C||f ||∞e−t∞∑k=0

tk

k!e−ck = C||f ||∞e−t(1−e

−c). (4.23)

Let us define a sequence of Markov semimartingales Ypn := n−

1α0 Y p

nt, n ∈ N. Now, weprove that the sequence of processes Yp

n, n ∈ N, converges in distribution to a symmetricα0-stable Levy process L with modified characteristics (relative to the truncation functionh(x))

B0t = Θt

∫R(h(y)− y1|y|≤1)

dy

|y|α0+1,

C0t = Θt

∫Rh2(y)

dy

|y|α0+1and

N0(ds, dy) = Θdyds

|y|α0+1

(see Theorem 2.5.16), where Θ :=∫R/Λ 1α(x)=α0c(x)π(dx). As mentioned in Theorem

2.6.3, we take all the processes Ypn, n ∈ N, and L to be defined on the same probability

spaces (Ω,F , Pxx∈R). In order to prove this convergence, by Theorem 2.6.3 it sufficesto show that initial distributions of Yp

n converge to initial distribution of L (which is

trivially satisfied) and the modified characteristics (Bn, Cn, Nn) of the processes Ypn, n ∈ N,

converge in probability to the modified characteristics (B0, C0, N0), when n −→ ∞. ByProposition 4.3.2 (iii), the modified characteristics (Bn, Cn, Nn) of the process Yp

n are given


by

Bnt = n

1+ 1α0

∫ t

0

∫Rh(y)fY pns

(n

1α0 y)dyds,

Cnt = n

1+ 1α0

∫ t

0

∫Rh2(y)fY pns

(n

1α0 y)dyds and

Nn(ds, dy) = n1+ 1

α0 fXpns

(n

1α0 y)dyds.

Note that (PC6), (3.1) and λ(x ∈ R : α(x) = α0 := infx∈R α(x)) > 0, i.e., π(ΠΛ(x ∈ R :α(x) = α0 := infx∈R α(x))) > 0, imply 0 < Θ < ∞, therefore the above α0-stable Levyprocess characteristics are well defined.

Let g ∈ Cb(R) vanish in a neighborhood of the origin. Using the definition of the∗-product (see (2.26)) we have

g ∗Nnt =

∫ t

0

∫Rg(y)Nn(ds, dy)

=

∫ t

0

∫Rg(y)n

1+ 1α0 fY pns

(n

1α0 y)dyds

=

∫ t

0

∫Rg(n

1

α(Ypns)− 1α0 y)n

1+ 1

α(Ypns)fY pns

(n

1

α(Ypns)y

)dyds

=

∫ t

0

∫R

1α(Xpns)=α0g

(n

1


1+ 1

α(Ypns)fY pns

(n

1

α(Ypns)y

)dyds

+

∫ t

0

∫R

1α(Y pns)>α0g(n

1


1+ 1

α(Ypns)fY pns

(n

1

α(Ypns)y

)dyds

=

∫ t

0

∫R

1α(Y pns)=α0g (y)c(Y p

ns)

|y|α0+1dyds (4.24)

+

∫ t

0

∫R

1α(Y pns)=α0g (y)

(n

1+ 1α0 fY pns

(n

1α0 y)− c(Y p

ns)

|y|α0+1

)dyds (4.25)

+

∫ t

0

∫R

1α(Y pns)>α0g (y)n1−α(Y

pns)α0

c(Y pns)

|y|α(Y pns)+1dyds (4.26)

+

∫ t

0

∫R

1α(Y pns)>α0g (y)

(n

1+ 1α0 fY pns

(n

1α0 y)− n1−α(Y

pns)α0

c(Y pns)

|y|α(Y pns)+1

)dyds.

(4.27)

Let 0 < ε < 1 be arbitrary. Then, by (PC5), there exists yε ≥ 1, such that

(1− ε) c(x)

|y|α(x)+1< fx(y) < (1 + ε)

c(x)

|y|α(x)+1(4.28)

holds for all |y| ≥ yε and all x ∈ R. Since the function g(x) vanishes in a neighborhood ofthe origin, by (4.28) and the dominated convergence theorem, (4.25) and (4.27) convergeto 0 Px-a.s., when n −→∞. Let us prove that (4.26) converges in L2(Ω,F ,Px) to 0, when


n −→∞. Define

Un(z) :=

∫Rg(y)

(1α(z)>α0n

1−α(z)α0

c(z)

|y|α(z)+1−∫R/Λ

1α(x)>α0n1−α(x)

α0c(x)

|y|α(x)+1π(dx)

)dy.

By τ -periodicity of the functions α(x) and c(x), the function Un(z) is τ -periodic and∫R/Λ

Un(z)π(dz) = 0.

Using Fubini’s theorem, Markov property and (4.23), we have

Ex[(∫ t

0

Un(Y pns)

)2]

= 2

∫ t

0

∫ s

0

Ex[Un(Y pns)Un(Y p

nr)]drds

= 2

∫ t

0

∫ s

0

Ex[Ex[Un(Y pns)|Fnr]Un(Y p

nr)]drds

= 2

∫ t

0

∫ s

0

Ex[Pn(s−r)Un(Y pnr)Un(Y p

nr)]drds

= 2

∫ t

0

∫ s

0

Ce−n(1−e−c)(s−r)||Un||2∞

=2C||Un||2∞n(1− e−c)

∫ t

0

(1− e−n(1−e−c)s)ds ≤ C||Un||2∞n(1− e−c)

. (4.29)

Note that ||Un||∞ remains bounded as n grows. Hence

limn−→∞

Ex[(∫ t

0

Un(Y pns)

)2]

= 0.

Furthermore,(Ex[(∫ t

0

∫R

1α(Y pns)>α0g (y)n1−α(Y

pns)α0

c(Y pns)

|y|α(Y pns)+1dyds

)2]) 1

2

≤

(Ex[(∫ t

0

Un(Y pns)

)2]) 1

2

+

(Ex[(∫ t

0

∫R

∫R/Λ

1α(x)>α0n1−α(x)

α0 g(y)c(x)

|y|α(x)+1π(dx)dy

)2]) 1

2

. (4.30)

By the dominated convergence theorem, (4.30) converges to zero, when n −→∞, i.e., (4.26)converges in L2(Ω,F ,Px) to 0, when n −→ ∞. Now, let us prove that (4.24) converges inL2(Ω,F ,Px) to

g ∗N0t = t

∫R

∫R/Λ

1α(x)=α0g(y)c(x)

|y|α0+1π(dx)dy,


when n −→∞. Define

U(z) :=

∫Rg(y)

(1α(z)=α0

c(z)

|y|α0+1−∫R/Λ

1α(x)=α0c(x)

|y|α0+1π(dx)

)dy.

By the τ -periodicity of the functions α(x) and c(x), the function U(z) is τ -periodic and∫R/Λ

U(z)π(dz) = 0.

Hence, in the same way as for (4.26), it can be shown that g ∗Nnt converges in probability

to g ∗ N0t . Also, in the same way, one can prove that Bn

t converges in probability to B0t ,

when n −→∞.At the end, let us show that Cn

t converges in probability to C0t , when n −→ ∞. Recall

that the truncation function h(x) is a bounded Borel measurable function satisfying h(x) =x in a neighborhood of the origin. Let δ > 0 be small enough and such that h(x) = x forall x ∈ (−δ, δ). We have

Cnt =

∫ t

0

∫Rh2(y)n

1+ 1α0 fY pns

(n

1α0 y)dyds

=

∫ t

0

∫R

1α(Y pns)=α0h2 (y)n

1+ 1α0 fY pns

(n

1α0 y)dyds

+

∫ t

0

∫R

1α(Y pns)>α0h2 (y)n

1+ 1α0 fY pns

(n

1α0 y)dyds

=

∫ t

0

∫R

1α(Y pns)=α0h2 (y)

c(Y pns)

|y|α0+1dyds (4.31)

+

∫ t

0

∫(−δ,δ)c

1α(Y pns)=α0h2 (y)

(n

1+ 1α0 fY pns

(n

1α0 y)− c(Y p

ns)

|y|α0+1

)dyds (4.32)

+

∫ t

0

∫(−δ,δ)

1α(Y pns)=α0y2

(n

1+ 1α0 fY pns

(n

1α0 y)− c(Y p

ns)

|y|α0+1

)dyds (4.33)

+

∫ t

0

∫R

1α(Y pns)>α0h2 (y)n

1−α(Ypns)α0

c(Y pns)

|y|α(Y pns)+1dyds (4.34)

+

∫ t

0

∫(−δ,δ)c

1α(Y pns)>α0h2 (y)

(n

1+ 1α0 fY pns

(n

1α0 y)− n1−α(Y

pns)α0

c(Y pns)

|y|α(Y pns)+1

)dyds

(4.35)

+

∫ t

0

∫(−δ,δ)

1α(Y pns)>α0y2

(n

1+ 1α0 fY pns

(n

1α0 y)− n1−α(Y

pns)α0

c(Y pns)

|y|α(Y pns)+1

)dyds. (4.36)

By (4.28) and the dominated convergence theorem, (4.32) and (4.35) converge to 0, Px-a.s.,when n −→∞. Let us prove that (4.33) converges to 0 Px-a.s., when n −→∞ and δ −→ 0,


respectively. By using (4.28), we have∫ t

0

∫(−δ,δ)

1α(Y pns)=α0y2

(n

1+ 1α0 fY pns

(n

1α0 y)− c(Y p

ns)

|y|α0+1

)dyds

=

∫ t

0

∫(−n

1α0 δ,n

1α0 δ)

1α(Y pns)=α0y2n

1− 2α0 fY pns (y) dyds

−∫ t

0

∫(−δ,δ)

1α(Y pns)=α0|y|1−α0c(Y p

ns)dyds

=

∫ t

0

∫(−yε,yε)

1α(Y pns)=α0y2n


+

∫ t

0

∫(−n

1α0 δ,−yε)∪(yε,n

1α0 δ)

1α(Y pns)=α0y2n


+2

2− α0

δ2−α0

∫ t

0

1α(Y pns)=α0c(Ypns)ds

≤ n1− 2

α0

∫ t

0

∫(−yε,yε)

1α(Y pns)=α0y2fY pns (y) dyds

+ (1 + ε)n1− 2

α0

∫ t

0

∫(−n

1α0 δ,−yε)∪(yε,n

1α0 δ)

1α(Y(n)s )=α0

y2 c(Ypns)

|y|α0+1dyds

+2

2− α0

δ2−α0

∫ t

0


= n1− 2

α0

∫ t

0

∫(−yε,yε)

1α(Y pns)=α0y2fY pns (y) dyds

+ (1 + ε)2

2− α0

δ2−α0

∫ t

0


− (1 + ε)n1− 2

α02

2− α0

y2−α0ε

∫ t

0


+2

2− α0

δ2−α0

∫ t

0

1α(Y pns)=α0c(Ypns)ds.

Now, by (3.1) and the dominated convergence theorem, we have

limδ−→0

limn−→∞

∫ t

0

∫(−δ,δ)

1α(Y pns)=α0y2

(n

1+ 1α0 fY pns

(n

1α0 y)− c(Y p

ns)

|y|α0+1

)dyds = 0 Px-a.s.

In completely the same way one can prove that (4.36) converges to 0, Px-a.s., when n −→∞and δ −→ 0, respectively. In order to prove that (4.34) converges in L2(Ω,F ,Px) to 0,when n −→∞, define

Vn(z) :=

∫Rh2(y)

(1α(z)>α0n

1−α(z)α0

c(z)

|y|α(z)+1−∫R/Λ

1α(x)>α0(x)n1−α(x)

α0c(x)

|y|α(x)+1π(dx)

)dy


and proceed as for (4.26). It remains to prove that (4.31) converges in L2(Ω,F ,Px) to C0t ,

when n −→∞. Let us define

V (z) :=

∫Rh2(y)

(1α(z)=α0

c(z)

|y|α0+1−∫R/Λ

1α(x)=α0c(x)

|y|α0+1π(dx)

)dy.

By τ -periodicity of the functions α(x) and c(x), the function V (z) is τ -periodic and∫R/Λ

V (z)π(dz) = 0.

Hence, by repeating the same computations as for (4.26), we have the claim. Therefore, byTheorem 2.6.3, we have proved that the sequence of processes Yp

n converges in distributionto the symmetric α0-stable Levy process L with the compensator (Levy measure) N0.

Now, let us prove that the stable-like chain Xp is recurrent if, and only if, α0 ≥ 1.By [15, Lemmas 2 and 3], the set of recurrent paths R(O) is a continuity set for theprobability measure PxL(·) for all x ∈ R and all open bounded sets O ⊆ R. Furthermore,since L is a λ-irreducible T-model (note that (2.28) is trivially satisfied), by Proposition2.7.12, L is recurrent if, and only if, PxL(R(O)) = 1 for all open bounded sets O ⊆ R andall x ∈ R, and it is transient if, and only if, PxL(T (O)) = 1 for all open bounded sets O ⊆ Rand all x ∈ R.

Let O ⊆ R be an arbitrary open bounded set and let x ∈ R be an arbitrary startingpoint. By Theorem 2.6.1, we have

limn−→∞

PxYpn(R(O)) = PxL(R(O)). (4.37)

If the stable-like chain Xp is recurrent, since it is λ-irreducible T-model, it is also H-recurrent. Hence, by Proposition 4.3.2 (ii), all the processes Yp

n, n ∈ N, are H-recurrent.This implies

PxYnn(R(O)) = 1 for all n ∈ N.

Therefore, by (4.37), PL(R(O)x) = 1, i.e., L is recurrent.Let us assume that the stable-like chain Xp is transient. Then, by Proposition 2.7.12,

Px (τO <∞) = 0 for all x ∈ R and all open bounded sets O ⊆ R. Hence, by Proposition4.3.2 (ii), Px (τnO <∞) = 0, i.e.,

PxYpn(R(O)) = 0

for all n ∈ N, all x ∈ R and all open bounded sets O ⊆ R. Therefore, by (4.37),PxL(R(O)) = 0, i.e., L is transient. Finally, by Corollary 2.7.14 (iii), we now that L isrecurrent if, and only if, α0 ≥ 1, which accomplishes the proof. 2

4.4 Recurrence and transience of (α, β)-stable-like Markov chains 83

4.4 Recurrence and transience of (α, β)-stable-like Markov

chains

In this section, we derive a recurrence and transience criterion for the (α, β)-stable-likeMarkov chain X(α,β). First, let us state the criterion for the continuous-time version of the(α, β)-stable-like chain X(α,β).

Theorem 4.4.1 [7, Corollary 5.5] Let k > 0, α, β ∈ (0, 2) and γ, δ ∈ (0,∞) be arbitrary.Furthermore, let α : R −→ (0, 2) and γ : R −→ (0,∞) be α, γ ∈ C1

b (R) and such that

α(x) =

α, x < −kβ, x > k

and γ(x) =

γ, x < −kδ, x > k.

Then, the stable-like process Y(α,β) determined by a symbol of the form p(x, ξ) = γ(x)|ξ|α(x)

is recurrent if, and only if, α + β ≥ 2. 2

In the discrete-time case we have a similar conclusion.

Theorem 4.4.2 The (α, β)-stable-like Markov chain X(α,β) is recurrent if, and only if,α + β ≥ 2. 2

The idea of the proof is to apply Theorem 2.6.5 and approximate the stable-like process

Y(α,β) by a sequence of Markov chains Yn, n ∈ N, such that X(α,β) d

= Y1 (recall that

the chain X(α,β)

is defined in Section 3.3, but in this case with the same stability functionα(x) and scaling function γ(x) as the stable-like process Y(α,β)), and to prove that allchains Yn, n ∈ N, are either recurrent or transient at the same time and that theirrecurrence property is equivalent with recurrence property of the stable-like process Y(α,β).By applying Proposition 2.7.11, this accomplishes the proof. We need some auxiliaryresults.

Lemma 4.4.3 Let 0 < ε < 2 and C > 0 be arbitrary, and let α : R −→ (ε, 2) andγ : R −→ (0, C) be arbitrary functions. Furthermore, let f(·;α(x), γ(x))x∈R be a familyof Sα(x)S density functions. Then, the following uniformity condition stands

limb−→∞

supx∈R

∫ ∞b

f(y;α(x), γ(x))dy = 0.

Moreover,

lim|y|−→∞

supx∈R:α(x)<1

∣∣∣∣f(y;α(x), γ(x))|y|α(x)+1

c(x)− 1

∣∣∣∣ = 0,

where

c(x) =

γ(x)

2, α(x) = 1

γ(x)π

Γ(α(x) + 1) sin(πα(x)

2

), α(x) 6= 1.


Proof. Let 0 < ρ < ε be arbitrary and let Zxx∈R be a family of random variables withSα(x)S distributions with density functions f(·;α(x), γ(x))x∈R. Then, we have

supx∈R

∫ ∞b

f(y;α(x), γ(x))dy = supx∈R

P(Zx ≥ b) ≤ supx∈R

P(|Zx| ≥ b) ≤ 1

bρsupx∈R

E|Zx|ρ.

Since supx∈R E|Zx|ρ is finite (see [40, page 163]), the first claim easily follows. The secondpart of lemma follows from Theorem 2.2.4 (iii). 2

By specifying the Markov chain Y from Proposition 4.3.2 (iii), we obtain the followingmore precise result.

Proposition 4.4.4 Let 0 < ε < 2 and C > 0 be arbitrary, let α : R −→ (ε, 2) andγ : R −→ (0, C) be continuous functions and let Xα(x) be a Markov chain given bytransition densities with following characteristic exponents

ψ(x; ξ) = γ(x)|ξ|α(x).

Furthermore, let a 6= 0 be arbitrary and let Nκ be a Poisson process with parameter κ > 0independent of the Markov chain Xα(x). Then, the process Yα(x) := aXα(x)

Nκtt∈R+ is

(i) a Feller process with symbol

p(x, ξ) = a−1κ

(1−

∫Reiξyf(a−1y;α(a−1x), γ(a−1x))dy

)and Levy triplet (0, 0, a−1κf(a−1y;α(a−1x), γ(a−1x))dy), and its corresponding semi-group satisfies the Cb-Feller property and strong Feller property

(ii) a T-model.

Proof. From Proposition 4.3.2 (i) we know that the semigroup of the process Yα(x) is givenby

Pα(x)t f(x) = e−κt

∞∑k=0

(κt)k

k!

∫Rpk(a−1x, a−1dy

)f(y),

for f ∈ Bb(R) and t ∈ R+, and the generator

Aα(x)f(x) = a−1κ

∫Rd

(f(y + x)− f(x))f(a−1y;α(a−1x), γ(a−1x))dy,

with the domain DAα(x) = Bb(R). Furthermore, it is shown that the semigroup is stronglycontinuous.

(i) The Cb-Feller property easily follows by Proposition 3.3.2, Theorem 2.7.8 (iii) and

Fatou’s lemma. Next, let us show that Pα(x)t (C0(R)) ⊆ C0(R) for all t ∈ R+. For

f ∈ C0(R), by the Cb-Feller property, Pα(x)t f ∈ Cb(R) for all t ∈ R+. It remains

to show that Pα(x)t f(x) vanishes at infinity for all f ∈ C0(R) and all t ∈ R+. Let


f ∈ C0(R) and ε > 0 be arbitrary such that ||f ||∞ < M , for some M ≥ 0. SinceCc(R) is dense in (C0(R), || · ||∞), there exists fε ∈ Cc(R) such that ||f − fε||∞ < ε.We have∣∣∣∣∫

Rp(a−1x, a−1dy)f(y)

∣∣∣∣ ≤ ∫Rp(a−1x, a−1dy)|f(y)| <

∫Rp(a−1x, a−1dy)|fε(y)|+ ε

= a−1

∫supp fε−x

f(a−1y;α(a−1x), γ(a−1x))|fε(y + x)|dy + ε

≤ a−1(M + ε)

∫supp fε−x

f(a−1y;α(a−1x), γ(a−1x))dy + ε.

Since supp fε is a compact set, by applying Lemma 4.4.3, the function x 7−→∫R p(a

−1x, a−1dy)f(y) is a C0(R) function. Now, by the dominated convergence the-orem we have the claim. Further, since the generator is given by

Aα(x)f(x) = a−1κ

∫Rd

(f(y + x)− f(x))f(a−1y;α(a−1x), γ(a−1x))dy,

with the domain DAα(x) = C0(R) (for Yα(x) as a Feller process), its symbol and Levytriplet are given by

p(x, ξ) = a−1κ

(1−

∫Reiξyf(a−1y;α(a−1x), γ(a−1x))dy

)and (0, 0, a−1κf(α(a−1x),γ(a−1x))(a

−1y)dy), respectively. The strong Feller property fol-lows from Theorem 2.4.10

(ii) The claim follows from Theorem 2.7.8 (ii). 2

By Corollary 2.7.14 (iii), the recurrence property of a SαS random walk, given by acharacteristic exponent ψ(ξ) = γ|ξ|α, depends only on the index of stability α ∈ (0, 2]and it does not depend on the scaling constant γ ∈ (0,∞). In the following propositionwe show that this is also the case with the chain Xα(x). We need the following technicallemma.

Lemma 4.4.5 Let αnn∈N ⊆ (0, 2) and γnn∈N ⊆ (0,∞) be two sequences with limitsα0 ∈ (0, 2) and γ0 ∈ (0,∞), respectively. Furthermore, let f(·;αn, γn)n∈N be a family ofSαnS density functions. Then, the family of density functions f(·;αn, γn)n∈N convergesuniformly on compact sets to the stable density function f(x;α0, γ0), i.e.,

limn−→∞

supx∈[a,b]

|f(x;αn, γn)− f(x;α0, γ0)| = 0

for all [a, b] ⊆ R.


Proof. First, by the dominated convergence theorem, we have the pointwise convergence

limn−→∞

f(x;αn, γn) = limn−→∞

1

2π

∫Re−iξxe−γn|ξ|

αndξ =

1

2π

∫Re−iξxe−γ0|ξ|

α0dξ = f(x;α0, γ0)

(4.38)

for all x ∈ R. Let ε > 0 and [a, b] be arbitrary. Without loss of generality let [a, b] ⊆ [0,∞).Since the density function f(x;α0, γ0) is continuous, it is uniformly continuous on [a, b].Therefore, there exists δ > 0 such that for all x, y ∈ [a, b], such that |x − y| < δ, we have|f(x;α0, γ0) − f(y;α0, γ0)| < ε

3. Let a = x0 < x1 < . . . < xp = b be a partition of the

segment [a, b], such that |xi − xi+1| < δ for all i = 0, ..., p − 1. Then, from (4.38), thereexists nε ∈ N, such that

|f(xi;αn, γn)− f(xi;α0, γ0)| < ε

3

holds for all n ≥ nε and all i = 0, ..., p. Let x ∈ [a, b] \ x0, ..., xp be arbitrary and let(xi, xi+1) be the unique interval, such that x ∈ (xi, xi+1). Then, for all n ≥ nε, we have

|f(x;αn, γn)− f(x;α0, γ0)| = |f(x;αn, γn)− f(xi+1;α0, γ0) + f(xi+1;α0, γ0)− f(x;α0, γ0)|

< |f(x;αn, γn)− f(xi+1;α0, γ0)|+ ε

3, (4.39)

and from Theorem 2.2.4 (vi) we have

f(xi+1;αn, γn)− f(xi+1;α0, γ0) < f(x;αn, γn)− f(xi+1;α0, γ0)

< f(xi;αn, γn)− f(xi+1;α0, γ0). (4.40)

Combining (4.39) and (4.40), for all n ≥ nε, we get

|f(x;αn, γn)− f(x;α0, γ0)|

< max|f(xi;αn, γn)− f(xi+1;α0, γ0)|, |f(xi+1;αn, γn)− f(xi+1;α0, γ0)|+ε

3

< max|f(xi;αn, γn)− f(xi;α0, γ0)|+ |f(xi;α0, γ0)− f(xi+1;α0, γ0)|, ε

3

+ε

3

< maxε

3+ε

3,ε

3

+ε

3= ε.

Hence,supx∈[a,b]

|f(x;αn, γn)− f(x;α0, γ0)| < ε

holds for all n ≥ nε. 2

Let c > 0 be arbitrary and let Xcα(x) be a Markov chain which we get by replacing thescaling function γ(x) by the scaling function cγ(x) in the chain Xα(x) defined in Proposition4.4.4

Proposition 4.4.6 The Markov chain Xcα(x) is recurrent if, and only if, the Markov chainXα(x) is recurrent.


Proof. Let N1 be a Poisson process with parameter κ = 1 independent of the chain Xcα(x).Define Xc := Xcα(x)

Ntt∈R+ . By Proposition 4.3.2 (i) and (iii) and Proposition 4.4.4 (i), the

process Xc is a Feller semimartingale with characteristics (relative to a truncation functionh(x)) of the form

Bct =

∫ t

0

∫R\0

h(y)f(y;α(Xcα(x)Ns

), cγ(Xcα(x)Ns

))dyds,

Cct = 0,

Cct =

∫ t

0

∫R\0

h2(y)f(y;α(Xcα(x)Ns

), cγ(Xcα(x)Ns

))dyds and

N c(ds, dy) = f(y;α(Xcα(x)Ns

), cγ(Xcα(x)Ns

))dyds.

Let c0 > 0 be arbitrary and fixed and let us show that

Xc ⇒ Xc0 , when c −→ c0.

We only have to check the assumptions from Theorem 2.6.4. Assumptions (i), (ii), (v),(vi) and (vii) can be easily verified, while assumption (iv) follows from Lemma 4.4.3. Toverify the assumption in (iii) we have to show that

limc−→c0

supx∈[a,b]

∣∣∣∣∫Rg(y) (f(y;α(x), c0γ(x))− f(y;α(x), cγ(x))) dy

∣∣∣∣ = 0

holds for all g ∈ Cb(R) and all [a, b] ⊆ R. If that would not be the case, then there wouldexist g ∈ Cb(R), [a, b] ⊆ R, δ > 0 and sequences cnn∈N ⊆ (0,∞) and xnn∈N ⊆ [a, b]with limits c0 and x0 ∈ [a, b], respectively, such that∣∣∣∣∫

Rg(y) (f(y;α(xn), c0γ(xn))− f(y;α(xn), cnγ(xn))) dy

∣∣∣∣ > δ (4.41)

holds for all n ∈ N. Let M ≥ 0 be such that ||g(x)||∞ ≤ M and let R > 0 be arbitrary.We have ∣∣∣∣∫

Rg(y) (f(y;α(xn), c0γ(xn))− f(y;α(xn), cnγ(xn))) dy

∣∣∣∣≤∣∣∣∣∫ R

−Rg(y) (f(y;α(xn), c0γ(xn))− f(y;α(xn), cnγ(xn))) dy

∣∣∣∣+

∣∣∣∣∫|y|≥R

g(y) (f(y;α(xn), c0γ(xn))− f(y;α(xn), cnγ(xn))) dy

∣∣∣∣ .By the continuity of the functions α(x) and γ(x) and Lemma 4.4.5, we have

limn−→∞

∣∣∣∣∫ R

−Rg(y) (f(y;α(xn), c0γ(xn))− f(y;α(xn), cnγ(xn))) dy

∣∣∣∣ = 0.


Furthermore, by Lemma 4.4.3 we have

limR−→∞

supn∈N

∣∣∣∣∫|y|≥R

g(y) (f(y;α(xn), c0γ(xn))− f(y;α(xn), cnγ(xn))) dy

∣∣∣∣≤M lim

R−→∞supn∈N

∫|y|≥R

f(y;α(xn), c0γ(xn)) +M limR−→∞

supn∈N

∫|y|≥R

f(y;α(xn), cnγ(xn)) = 0.

Hence,

limn−→∞

∣∣∣∣∫Rg(y) (f(y;α(xn), c0γ(xn))− f(y;α(xn), cnγ(xn))) dy

∣∣∣∣ = 0,

what is in contradiction with (4.41). The locally uniform convergence of other two char-acteristics can be shown in a similar way.

Let x ∈ R be arbitrary and let O ⊆ R be an arbitrary open bounded set. Since theprocess Xc0 is a Cb-Feller process, by [15, Lemmas 2 and 3], R(O) is a continuity set of theprobability measures PxXc0 (·), x ∈ R. Therefore, by Theorem 2.6.1, we have

limc−→c0

PxXc(R(O)) = PxXc0 (R(O))

for all x ∈ R and all open bounded sets O ⊆ R. Hence, for all x ∈ R and all open boundedsets O ⊆ R, the function

c 7−→ PxXc(R(O))

is a continuous function on (0,∞). Since Xc is a λ-irreducible T-model, by Proposition2.7.12, PxXc(R(O)) = 1 for all c ∈ (0,∞), all x ∈ R and all open bounded sets O ⊆ R,or PxXc(R(O)) = 0 for all c ∈ (0,∞), all x ∈ R and all open bounded sets O ⊆ R. Notethat (2.28) is satisfied if for an arbitrary petite set C ∈ B(R), for the distribution aC(·)we take aC(·) := δt0(·), where t0 > 0 is arbitrary. Hence, again by Proposition 2.7.12, allthe processes Xc, c ∈ (0,∞), are either recurrent or transient at the same time. Now, byProposition 4.3.2 (ii), the desired result follows. 2

Now, we are ready to prove Theorem 4.4.2.

Proof of Theorem 4.4.2. Let α(x) and γ(x) be as in Theorem 4.4.1. According to

Proposition 2.7.11, it suffices to prove that the chain X(α,β)

is recurrent if, and only if,α + β ≥ 2.

Let Y(α,β) be a stable-like process determined by a symbol of the form p(x, ξ) =

γ(x)|ξ|α(x). By [24, Theorem 5.1], the transition function Px(Y (α,β)t ∈ dy) is absolutely

continuous with respect to the Lebesgue measure and Px(Y (α,β)t ∈ B) > 0 holds for all

x ∈ R, all t ∈ R+ and all B ∈ B(R) with λ(B) > 0. Hence, the stable-like process Y(α,β)

is λ-irreducible and, since its corresponding semigroup satisfies the Cb-Feller property, byTheorem 2.7.8 (ii) it is a T-model. Hence, by Proposition 2.7.10, it is either H-recurrentor transient. Furthermore, by Theorem 4.4.1, the stable-like process Y(α,β) is recurrent if,and only if, α + β ≥ 2.

By [2, Corollary 2.3] and [43, Proposition 4.6], C∞c (R) is an operator core for the infinites-imal generator of the stable-like process Y(α,β). Hence, by Theorem 2.6.5, the stable-like


process Y(α,β) can be approximated by a sequence of Markov chains Ym, m ∈ N, given bya sequence of transition kernels pm(x, dy), m ∈ N, such that∫

Reiξypm(x, dy) = eiξx−

γ(x)m|ξ|α(x) ,

i.e.,Ym ⇒ Y(α,β), as n −→∞,

where Yn

= Y mbmtct∈R+ . By Proposition 4.4.6, the chains Ym, m ∈ N, are either recurrent

or transient at the same time. The rest of the proof is devoted to proving that thisdichotomy is equivalent with the recurrence transience dichotomy of the stable-like processY(α,β).

Since the stable-like process Y(α,β) is a Cb-Feller process, by [15, Lemmas 2 and 3] wehave Px

Y(α,β)(∂R(O)) = 0 for all x ∈ R and all open bounded sets O ⊆ R. Therefore, byTheorem 2.6.1, we have

limm−→∞

PxYm(R(O)) = PxY(α,β)(R(O)) (4.42)

for all x ∈ R and for all open bounded sets O ⊆ R.Let us assume that α + β ≥ 2. Hence, the stable-like process Y(α,β) is recurrent. Note

that assumption (2.28), for the stable-like process Y(α,β), follows if for the distributionaC(·) we take aC(·) := δt0(·), where t0 > 0 is arbitrary, and use the strong Feller property.Therefore, by Proposition 2.7.12, Px

Y(α,β)(R(O)) = 1 holds for all x ∈ R and all openbounded sets O ⊆ R. From (4.42), for any starting point x ∈ R and any open bounded set

O ⊆ R there exists m0 ≥ 1 such that PxYm0 (R(O)) > 0, i.e., Px

(∑∞n=0 1Xm0

n ∈O =∞)> 0.

But, since the chain Ym0 is a λ-irreducible T-model, by Proposition 2.7.12, we have

Px(∞∑n=0

1Ym0n ∈O =∞

)= 1

for all x ∈ R and all open bounded sets O ⊆ R, i.e., the chain Ym0 is recurrent. Now,by applying Proposition 4.4.6, all the chains Ym, m ∈ N, are recurrent. Therefore, since

X(α,β) d

= Y1, by Propositions 2.7.11 the stable-like chain X(α,β) is recurrent.Now, let us show that the recurrence property of the stable-like chain X(α,β) implies

α + β ≥ 2. Let us assume that this is not the case, i.e., let us assume that α + β < 2.Hence, the stable-like process Y(α,β) is transient, i.e., Px

Y(α,β)(T (O)) = 1 holds for all x ∈ Rand all open bounded sets O ⊆ R. Now, by (4.42), we have

limm−→∞

PxYm(T (O)) = PxY(α,β)(T (O)) = 1.

Recall that T (O) = R(O)c. Hence, for any starting point x ∈ R and any openbounded set O ⊆ R, there exists m0 ≥ 1 such that PxYm0 (T (O)) > 0. Therefore,


Px(∑∞

n=0 1Ym0n ∈O =∞

)< 1. Again, by Proposition 2.7.12, we have

Px(∞∑n=0

1Ym0n ∈O =∞

)= 0

for all x ∈ R and all open bounded sets O ⊆ R. Hence, the chain Ym0 is transient. There-

fore, by Proposition 4.4.6, all the chains Ym, m ∈ N, are transient. Since X(α,β) d

= Y1, byProposition 2.7.11, the stable-like chain Y(α,β) is also transient. But this is in contradictionwith the recurrence assumption of the stable-like chain X(α,β). Hence, we have proved thedesired result. 2

Remark 4.4.7 As already mentioned, it is shown in [15] that if the functions α : R −→(0, 2) and γ : R −→ (0,∞) are continuously differentiable and periodic and if the setx ∈ R : α(x) = α0 := infx∈R α(x) has positive Lebesgue measure, then the stable-likeprocess with symbol p(x, ξ) = γ(x)|ξ|α(x) is recurrent if, and only if, α0 ≥ 1. In general, wecannot apply Theorem 4.3.1 for the discrete-time version of this stable-like process, since itstransition densities do not satisfy Proposition 3.3.1. But, by repeating the proof of Theorem4.4.2 we deduce the following. If α : R −→ (0, 2) and γ : R −→ (0,∞) are continuouslydifferentiable and periodic functions and if the set x ∈ R : α(x) = α0 := infx∈R α(x)has positive Lebesgue measure, then the Markov chain given by transition densities withfollowing characteristic exponents

ψ(x; ξ) = γ(x)|ξ|α(x)

is recurrent if, and only if, α0 ≥ 1.Similarly, by repeating the proof of Theorem 4.4.2, we can prove the transience property

of the discrete-time version of the stable-like process considered in [43], i.e., the processgiven by symbol p(x, ξ) = γ(x)|ξ|α(x), where α : R −→ (0, 2) and γ : R −→ (0,∞) are ofclass C1

b (R) and such that 0 < infx∈R α(x) ≤ lim sup|x|−→∞ α(x) < 1 and 0 < infx∈R γ(x).Note that this case is actually covered by Theorem 4.2.1.

4.5 Applications and generalizations

In this section, we discuss several consequences and generalizations of the previousresults. As a simple consequence of Theorems 4.1.1 and 4.2.1 we get a new proof ofCorollary 2.7.14 (iii).

Corollary 4.5.1 A SαS, 1 < α < 2, random walk is recurrent. A Sα(γ, δ), 0 < α < 1,random walk with arbitrary shift is transient. 2

Theorem 4.4.2 generalize previous corollary.

Corollary 4.5.2 A SαS, 0 < α < 2, random walk is recurrent if, and only if, α ≥ 1. 2

4.5 Applications and generalizations 91

If we apply Theorems 4.1.1 and 4.2.1 in the general random walk case, i.e., in the casewhen the stable-like chain X has jump density function f(y) with f(y) ∼ c|y|−α−1, when|y| −→ ∞, where α ∈ (0, 2) and c ∈ (0,∞), (note that in this case conditions (C1)-(C5)are trivially satisfied) then, by Theorem 4.1.1 and (4.3), if α > 1 and if∫

Ry f(y)dy = 0,

the stable-like chain X is recurrent, and if α < 1, by Theorem 4.2.1 and (4.16), the stable-like chain X is transient. This result can be strengthened. If, in addition, we assumethat f ∈ C(R), f(y) = f(−y) and f(y) > 0 for all y ∈ R, by Theorem 4.3.1, we get thefollowing.

Corollary 4.5.3 The periodic stable-like chain Xp with jump density function f(y) is re-current if, and only if, α ≥ 1. 2

We also give an alternative proof.

Proposition 4.5.4 [45, Discussion on page 88] A random walk with jump density functionf(y), with f(y) = f(−y) for all y ∈ R, and f(y) ∼ c|y|−α−1, when |y| −→ ∞, whereα ∈ (0, 2) and c ∈ (0,∞), is recurrent if, and only if, α ≥ 1.

Proof. Let ϕ(ξ) be the characteristic function of the density function f(y). We have

1− ϕ(ξ)

|ξ|α=

∫R|y|1+α|ξ|f(y)

1− cos(ξy)

|ξy|1+αdy.

Define a function g : R −→ R by g(y) := 1−cos y|y|1+α . We have

1− ϕ(ξ)

|ξ|α= c

∫R|ξ|g(ξy)dy +

∫R|ξ|g(ξy)ε(y)dy,

whereε(y) = |y|1+αf(y)− c.

Recall that c ∈ (0,∞) is such that f(y) ∼ c|y|−α−1, when |y| −→ ∞. Hence,

lim|y|−→∞

ε(y) = 0

and since 1− cos y ∼ y2

2, when y −→ 0,∫

R|ξ|g(ξy)dy =

∫R

1− cos y

|y|α+1dy <∞

for all 0 < α < 2. Put M :=

∫ ∞−∞

1− cos y

|y|α+1dy > 0 and take ε > 0 arbitrary. Then, there

exists yε > 0, such that for all |y| ≥ yε we have |ε(y)| < εM. Hence, by the dominated


convergence theorem and the fact 1− cos y ∼ y2

2, when y −→ 0, we have

limξ−→0

∣∣∣∣∫R|ξ|g(ξy)ε(y)dy

∣∣∣∣ ≤ limξ−→0

∫ yε

−yε|ξ|g(ξy)|ε(y)|dy +

ε

Mlimξ−→0

∫(−yε,yε)c

|ξ|g(ξy)dy < ε,

i.e.,

limξ−→0

∫R|ξ|g(ξy)ε(y) = 0.

Therefore,

limξ−→0

1− ϕ(ξ)

|ξ|α= cM.

Now, the claim follows from Theorem 2.7.13 (iv). 2

Let us give two examples of applications of Theorems 4.1.1, 4.2.1 and 4.3.1.

Example 4.5.5 Let α1, . . . , αr ∈ (0, 2), γ1, . . . , γs ∈ (0,∞) and δ1, . . . , δt ∈ R be arbitrary.Furthermore, let A1, . . . , Ar, C1, . . . , Cs, D1, . . . , Dt ∈ B(R) be arbitrary sets such that

(i) Ai ∩ Aj = ∅, Ci ∩ Cj = ∅ and Di ∩Dj = ∅, for i 6= j

(ii) ∪ri=1Ai = R, ∪si=1Ci = R and ∪ti=1Di = R.

Let us define α(x) :=∑r

i=1 αi1Ai(x), γ(x) :=∑s

i=1 γi1Ci(x) and δ(x) :=∑t

i=1 δi1Di(x).Then, by Proposition 3.3.1, the family of stable density functions f(·;α(x), γ(x), δ(x))x∈Rsatisfies conditions (C1)-(C5).

If α1, . . . , αr ∈ (1, 2) and δ1 = . . . = δu = 0, then, from Theorem 4.1.1, the stable-like chain X given by the transition function p(x, dy) = f(y − x;α(x), γ(x), δ(x))dy isrecurrent. If α1, . . . , αr ∈ (0, 1) then, from Theorem 4.2.1, the stable-like chain X given bythe transition function p(x, dy) = f(y − x;α(x), γ(x), δ(x))dy is transient.

Example 4.5.6 Let α : R −→ (0, 2) be an arbitrary continuous periodic function withperiod τ > 0. Define a family of density functions fxx∈R on R by

fx(y) :=

12

α(x)α(x)+1

, |y| ≤ 112

α(x)α(x)+1

|y|−α(x)−1, |y| ≥ 1

for all x ∈ R. It is easy to verify that the family of density functions fxx∈R satisfiesconditions (PC1)-(PC6). Hence, if the set x ∈ R : α(x) = α0 := infx∈R α(x) haspositive Lebesgue measure, by Theorem 4.3.1, the periodic stable-like chain Xp given bythe transition function p(x, dy) = fx(y − x)dy is recurrent if, and only if, α0 ≥ 1.

In Section 3.1 we commented that a change of the stable-like chain X on a set of Lebesguemeasure zero or on a bounded set will not change its recurrence or transience property.Hence, we can weaken assumptions on the stability function α(x) and the conditions in(4.1) and (4.15) in Theorems 4.1.1 and 4.2.1, respectively. In Theorem 4.1.1 we assumedthat α : R −→ (0, 2) and

lim inf|x|−→∞

α(x) > 1,

4.5 Applications and generalizations 93

but it is enough to request that α : R \ (A ∪B) −→ (1, 2) and

lim infx∈R\A, |x|−→∞

α(x) > 1,

for some set A ∈ B(R) with zero Lebesgue measure and some bounded set B ∈ B(R). In(4.1), instead of using lim sup|x|−→∞, we use lim supx∈R\A, |x|−→∞. An analog modificationcan be done in Theorem 4.2.1.

Theorems 4.1.1 and 4.2.1 can also be generalized in the spirit of Remark 3.1.9. Wemove from asymptotically symmetric transition jumps to asymptotically non-symmetrictransition jumps, i.e., we replace the stability function α(x) and scaling function γ(x),which satisfy conditions (C1)-(C5), by two stability functions α−(x) and α+(x) and two

scaling functions c−(x) and c+(x), which satisfy conditions (C1)-(C5). This stable-likeMarkov chain we denoted by X. By repeating the proof of Theorem 4.1.1, the stable-likechain X is recurrent if α+, α− : R −→ (0, 2) are such that

lim|x|−→∞

α+(x)

α−(x)= 1 and α := lim inf

|x|−→∞α+(x)

(= lim inf|x|−→∞

α−(x)

)> 1,

and c+, c− : R −→ (0,∞) are such that

lim|x|−→∞

c−(x)

c+(x)|x|α+(x)−α−(x) = 1

and such that condition (4.1) is satisfied with the constant R(α). In this case, for a testfunction V (x), we take V (x) = ln(1 + |x|) again. Similarly, the stable-like chain X istransient if α+, α− : R −→ (0, 2) are such that

α+ := lim sup|x|−→∞

α+(x) < 1 and α− := lim sup|x|−→∞

α−(x) < 1,

and c+, c− : R −→ (0,∞) are such that

lim|x|−→∞

α+(x)c−(x)

c+(x)α−(x)|x|α+(x)−α−(x) = 1

and such that condition (4.15) is satisfied with the constant T (α, β), where α := α− ∨ α+

and β ∈ (0, 1 − α), and for some a0 > 0. In this case, for a test function V (x), we take

V (x) = 1− (1 + |x|)−β again.The above generalization is not too strong, i.e., the stability functions α−(x) and α+(x)

and the scaling functions c−(x) and c+(x) must be close to each other. The reason for thatare the chosen test functions V (x). Both, in the recurrence and transience case, the testfunctions V (x) are symmetric functions. Hence, we can treat the symmetric case only.

By the same arguments as in the proof of Theorem 4.3.1, we can deduce the recurrenceand transience properties of the periodic stable-like chain with non-symmetric transitionjumps X

p, described in Remark 3.2.1. That is, if the set x ∈ R : α(x) = α0 := infx∈R α(x)

has positive Lebesgue measure, then by subordination of the stable-like chain Xp

with a


Poisson process N1 with parameter κ = 1 (independent of Xp), one can prove that the

sequence of processes Yp

n := n−1α0 Xp

Nntt∈R+ , n ∈ N, converges in distribution to α0-

stable Levy process. In general, this α0-stable Levy process is not symmetric anymore.Non-symmetry of the corresponding densities fxx∈R implies that the α0-stable Levyprocess has a nonzero shift parameter, and two-tail behavior implies that the α0-stableLevy process has a nonzero skewness parameter. Hence, by Corollary 2.7.14 (iii), the onlyrecurrent cases are when either α0 > 1 and the shift parameter vanishes or α0 = 1 and theskewness parameter vanishes.

Chapter 5

Discrete state case

In this chapter, we derive the same recurrence and transience conditions for discreteanalogues of stable-like Markov chains X, Xp and X(α,β) as in Theorems 4.1.1, 4.2.1,4.3.1 and 4.4.2. Without loss of generality, we treat the case on the state space Z. Letα : Z −→ (0, 2) and c : Z −→ (0,∞) be arbitrary functions and let fii∈Z be a family ofprobability functions on Z which satisfies

(DC1) fi(j) ∼ c(i)|j|−α(i)−1, when |j| −→ ∞, for all i ∈ Z.

Let Xd be a Markov chain on Z given by the following transition kernel

p(i, j) := fi(j − i).

It is clear that if fi(j) > 0 for all i, j ∈ Z, then the chain Xd is irreducible. In general,this is not the case. We introduce a discrete version of conditions (C1)-(C5). Note thatconditions (C1)-(C5), in the discrete state case, are reduced just to conditions (C2) and(C3), since compact sets are replaced by finite sets. Therefore, we assume that the familyof probability functions fii∈Z satisfies the following condition

(DC2) there exists k ∈ N such that

lim|j|−→∞

supi∈−k,...,kc

∣∣∣∣fi(j) |j|α(i)+1

c(i)− 1

∣∣∣∣ = 0.

In the sequel we assume that the chain Xd satisfies conditions (DC1) and (DC2). It is easyto see that the chain Xd is

(i) irreducible (the irreducibility measure is the counting measure)

(ii) aperiodic, since p2(i, i) > 0 and p3(i, i) > 0 for all i ∈ Z

(iii) either recurrent or transient (see Theorem 2.7.3).

Let us consider now two special cases of the chain Xd, the discrete analogues of stable-like chains Xp and X(α,β).

95

96 5. Discrete state case

(i) Let m ≥ 1, α0, . . . , αm−1 ∈ (0, 2) and c0, . . . , cm−1 ∈ (0,∞) be arbitrary. Let Xdp bea Markov chain on Z given by

α(i) = αj and c(i) = cj

for i ≡ jmod (m), i.e., the functions α : Z −→ (0, 2) and c : Z −→ (0,∞) areperiodic functions with period m.

(ii) Let α, β ∈ (0, 2) and c, d ∈ (0,∞) be arbitrary. Let Xd(α,β) be a Markov chain on Zgiven by

α(i) =

α, i < 0β, i ≥ 0

and c(i) =

c, i < 0d, i ≥ 0.

5.1 Recurrence and transience of general discrete

stable-like Markov chains

Due to the tail behavior of transition jumps (transition functions are power functions)we can switch from sums to integrals. Hence, by using Foster-Lypunov drift conditionswith the same test functions and repeating the proofs of Theorems 4.1.1 and 4.2.1, we havethe following.

Theorem 5.1.1 Let α : Z −→ (0, 2) be an arbitrary function such that

α := lim inf|i|−→∞

α(i) > 1.

Furthermore, let c : Z −→ (0,∞) be an arbitrary function and let fii∈Z be a family ofprobability functions on Z which satisfies conditions (DC1) and (DC2) and such that

lim supδ−→0

lim sup|i|−→∞

(1 + |i|)α(i)

c(i)

bδ(1+|i|)c∑j=−bδ(1+|i|)c

ln

(1 + sgn (i)

j

1 + |i |

)fi(j) < R(α) (5.1)

when α < 2, where

R(α) :=∞∑k=1

1

k(2k − α)− ln 2

α− 1

2α

(Ψ

(α + 1

2

)−Ψ

(α2

)),

and the left-hand side in relation (5.1) is finite when α = 2. Then, the chain Xd is recur-rent. 2

Theorem 5.1.2 Let α : Z −→ (0, 2) be an arbitrary function such that

α := lim sup|i|−→∞

α(i) < 1

5.1 Recurrence and transience of general discrete stable-like Markov chains 97

and let β ∈ (0, 1 − α) be arbitrary. Furthermore, let c : Z −→ (0,∞) be an arbitraryfunction and let fii∈Z be a family of probability functions which satisfies conditions(DC1) and (DC2) and there exists a0 ∈ N, such that

lim inf|i|−→∞

α(i)|i|α(i)

c(i)

a∑j=−a

(1−

(1 + sgn (i)

j

1 + |i |

)−β)fi(j) > −T (α, β) (5.2)

for all a ≥ a0, a ∈ N, where

T (α, β) := 2F1(−α, β, 1− α; 1) + βB(1;α + β, 1− α)− αB(1;α + β, 1− β).

Then, the chain Xd is transient. 2

Using completely the same arguments and proofs as in the continuous state space case,by switching from sums to integrals, conditions (5.1) and (5.2) can be weakened to theconditions

lim sup|i|−→∞

sgn (i)|i |α(i)−1

c(i)Ei [X d

1 − i ] < R(α) (5.3)

and

lim sup|i|−→∞

α(i)

c(i)|i|α(i)−1 <

T (α, β)

a0β. (5.4)

If we apply Theorems 5.1.1 and 5.1.2 in the random walk case, i.e., in the case when thechain Xd has jump probability function f(j) with f(j) ∼ c|j|−α−1, when |j| −→ ∞, whereα ∈ (0, 2) and c ∈ (0,∞), then, by Theorem 5.1.1 and condition (5.3), if α > 1 and if∑

j∈Z

j f(j) = 0,

the chain Xd is recurrent, and if α < 1, by Theorem 5.1.2 and condition (5.4), the chainXd is transient. This result can be strengthened. Recalling the proof of Proposition 4.5.4we get the following.

Proposition 5.1.3 [45, Discussion on page 88] A random walk with jump probabilityfunction f(j), satisfying f(j) = f(−j) for all j ∈ Z, and f(j) ∼ c|j|−α−1, when |j| −→ ∞,where α ∈ (0, 2) and c ∈ (0,∞), is recurrent if, and only if, α ≥ 1.

Proof. Let ϕ(ξ) be the characteristic function of the probability function f(j). We have

1− ϕ(ξ)

|ξ|α=∑j∈Z

|j|1+α|ξ|f(j)1− cos(ξj)

|ξj|1+α.


Define a function g : R −→ R by g(y) := 1−cos y|y|1+α . We have

1− ϕ(ξ)

|ξ|α= c

∑j∈Z

|ξ|g(ξj) +∑j∈Z

|ξ|g(ξj)ε(j),

whereε(j) = |j|1+αf(j)− c.

Recall that c ∈ (0,∞) such that f(j) ∼ c|j|−α−1, when |j| −→ ∞. Hence,

lim|j|−→∞

ε(j) = 0

and since 1− cos y ∼ y2

2, when y −→ 0,

limξ−→0

∑j∈Z

|ξ|g(ξj) =

∫Rg(y)dy =

∫R

1− cos y

|y|α+1dy <∞

for all 0 < α < 2. Put M :=

∫R

1− cos y

|y|α+1> 0 and take ε > 0 arbitrary. Then, there exists

jε ∈ N, such that for all |j| ≥ jε we have |ε(j)| < εM. Hence, by the dominated convergence

theorem and the fact 1− cos y ∼ y2

2, when y −→ 0, we have

limξ−→0

∣∣∣∣∣∑j∈Z

|ξ|g(ξj)ε(j)

∣∣∣∣∣ ≤ limξ−→0

jε−1∑j=−jε+1

|ξ|g(ξj)|ε(j)|+ ε

Mlimξ−→0

∑|j|≥jε

|ξ|g(ξj) < ε,

i.e.,

limξ−→0

∑j∈Z

|ξ|g(ξj)ε(j) = 0.

Therefore,

limξ−→0

1− ϕ(ξ)

|ξ|α= cM.

Now, the claim follows from Theorem 2.7.13 (iv). 2

5.2 Recurrence and transience of discrete periodic

stable-like Markov chains

In this section, we derive recurrence and transience conditions for the chain Xdp. First,let us assume that m = 2, i.e., the stability function and scaling function are given by

α(i) =

α, i ∈ 2Zβ, i ∈ 2Z + 1

and c(i) =

c, i ∈ 2Zd, i ∈ 2Z + 1,

5.2 Recurrence and transience of discrete periodic stable-like Markov chains 99

respectively. Furthermore, assume that the probability functions fii∈Z satisfy fi(−j) =fi(j) for all i, j ∈ Z. Denote fi(j) by f(α,c)(j) if i ∈ 2Z and by f(β,d)(j) if i ∈ 2Z + 1.

Further, let us define the following stopping times inductively, Tα0 := 0, T β0 := 0,Tαn := infk > Tαn−1 : Xdp

k ∈ 2Z and T βn := infk > T βn−1 : Xdpk ∈ 2Z + 1, for n ∈ N.

Proposition 5.2.1 We have Pi(Tαn <∞) = Pi(T βn <∞) = 1 for all i ∈ Z and all n ∈ N.

Proof. Let us prove that Pi(Tαn < ∞) = 1 for all i ∈ Z and all n ∈ N by induction. Leti ∈ Z be arbitrary and let n = 1. We have

Pi(Tα1 =∞) =Pi(Xpdk ∈ 2Z + 1, ∀k ∈ N) = lim

k−→∞Pi(Xdp

l ∈ 2Z + 1, 1 ≤ l ≤ k)

= limk−→∞

∑i1∈2Z+1

p(i, i1)∑

i2∈2Z+1

p(i1, i2) . . .∑

ik−1∈2Z+1

p(ik−2, ik−1)p(ik−1, 2Z + 1).

Note that p(2i + 1, 2Z + 1) =∑

j∈2Z f(β,d)(j) < 1 for all i ∈ Z. Therefore, if we putC :=

∑j∈2Z f(β,d)(j) and Ci := p(i, 2Z + 1), we have

Pi(Tα1 =∞) = limk−→∞

CiCk−1 = 0,

i.e., Pi(Tα1 < ∞) = 1. Let us assume that Pi(Tαn−1 < ∞) = 1 and let us prove thatPi(Tαn <∞) = 1. By denoting N := Tαn−1 and using strong Markov property we have

Pi(Tαn <∞) = Ei[Ei[1Tα1 <∞ θN |FN ]] = Ei[EXN [1Tα1 <∞]] =∑j∈2Z

Ei[1XN=j] = 1.

In the completely analogous way we prove that Pi(T βn <∞) = 1 for all i ∈ Z and all n ∈ N,which proves the assertion. 2

According to the previous proposition, Yα := XdpTαnn∈Z+ and Yβ := Xdp

Tβnn∈Z+ are

well defined Markov chains, on the same probability space as the chain Xdp, with statespaces 2Z and 2Z + 1, respectively. For i ∈ Z, define the following stopping times: τi :=infn ≥ 1 : Xdp

n = i, ταi := infn ≥ 1 : Y αn = i and τβi = infn ≥ 1 : Y β

n = i.

Proposition 5.2.2 For all i ∈ Z, n ∈ N, j1, . . . , jn ∈ 2Z and all k1, . . . , kn ∈ 2Z + 1 wehave Pi(Y α

1 = j1, . . . , Yαn = jn) > 0 and Pi(Y β

1 = k1, . . . , Yβn = kn) > 0. In particular, the

Markov chains Yα and Yβ are irreducible on their state spaces.

Proof. Let i ∈ Z and j1 ∈ 2Z be arbitrary, then we have

Pi(Y α1 = j1) =p(i, j1) +

∑i1∈2Z+1

p(i, i1)p(i1, j1) +∑

i1∈2Z+1

p(i, i1)∑

i2∈2Z+1

p(i1, i2)p(i2, j1) + . . .

≥∑

i1∈2Z+1

p(i, i1)p(i1, j1).

If i ∈ 2Z, then we take i1 ∈ 2Z + 1 such that f(α,c)(i1 − i) > 0 and f(β,d)(j1 − i1) > 0.


Therefore,Pi(Y α

1 = j1) ≥ f(α,c)(i1 − i)f(β,d)(j1 − i1) > 0.

If i ∈ 2Z + 1, then we take i1 ∈ 2Z + 1 such that f(β,d)(i1 − i) > 0 and f(β,d)(j1 − i1) > 0.Hence, we have

Pi(Y α1 = j1) ≥ f(β,d)(i1 − i)f(β,d)(j1 − i1) > 0.

Let i ∈ Z and j1, j2 ∈ 2Z be arbitrary, then we have

Pi(Y α1 = j1, Y

α2 = j2) = Pi(Y α

2 = j2|Y α1 = j1)Pi(Y α

1 = j1) = Pj1(Y α1 = j2)Pi(Y α

1 = j1) > 0.

Let n > 2. Let us suppose that for all i ∈ Z and for all j1, . . . jn−1 ∈ 2Z we have

Pi(Y α1 = j1, . . . , Y

αn−1 = jn−1) > 0.

Let jn ∈ 2Z be arbitrary, then we have

Pi(Y α1 = j1, . . . , Y

αn = jn)

= Pi(Y αn = jn|Y α

n−1 = jn−1, . . . , Yα

1 = j1)Pi(Y α1 = j1, . . . , Y

αn−1 = jn−1)

= Pjn−1(Y α1 = jn)Pi(Y α

1 = j1, . . . , Yαn−1 = jn−1) > 0.

Analogously we prove the claim for the chain Yβ. Let i, j ∈ 2Z be arbitrary, then we have

Pi(ταj <∞) ≥ Pi(ταj = 1) = Pi(Y α1 = j) > 0.

Similarly, for arbitrary i, j ∈ 2Z + 1 we have

Pi(τβj <∞) > 0.

Hence, the chains Yα and Yβ are irreducible. 2

Proposition 5.2.3 The chains Xdp, Yα and Yβ are recurrent (respectively, transient) atthe same time.

Proof. Let i ∈ 2Z be arbitrary, then we have

Pi(ταi =∞) = Pi(Y αn ∈ 2Z \ i, n ∈ N) = Pi(Xdp

n ∈ Z \ i, n ∈ N) = Pi(τi =∞).

Similarly, for arbitrary i ∈ 2Z + 1 we have Pi(τi =∞) = Pi(τβi =∞). 2

Proposition 5.2.4 The chains Yα and Yβ are symmetric random walks with jump dis-tributions P0(Y α

1 ∈ ·) and P1(Y β1 − 1 ∈ ·), respectively.

5.2 Recurrence and transience of discrete periodic stable-like Markov chains 101

Proof. First, note that for arbitrary i, j ∈ Z we have

P0(Y αn+1 = 2i− 2j|Y α

n = 0)

= p(0, 2i− 2j) +∑

k1∈2Z+1

p(0, k1)p(k1, 2i− 2j)

+∑

k1∈2Z+1

p(0, k1)∑

k2∈2Z+1

p(k1, k2)p(k2, 2i− 2j) + . . .

= p(2i, 2j) +∑

k1∈2Z+1

p(2j, k1 + 2j)p(k1 + 2j, 2i)

+∑

k1∈2Z+1

p(2j, k1 + 2j)∑

k2∈2Z+1

p(k1 + 2j, k2 + 2j)p(k2 + 2j, 2i) + . . .

= P0(Y αn+1 = 2i|Y α

n = 2j).

Let us prove that the random variables Y αn+1 − Y α

n , n ≥ 0, are symmetric i.i.d. randomvariables with respect to the probability measure P0(·). Let n ≥ 0. Then, we have

P0(Y αn+1 − Y α

n = 2i) =∑j∈Z

P0(Y αn+1 = 2i+ 2j, Y α

n = 2j)

=∑j∈Z

P0(Y αn+1 = 2i+ 2j|Y α

n = 2j)P0(Y αn = 2j) = P0(Y α

1 = 2i).

Let n ≥ 1. Then, we have

P0(Y αn+1 − Y α

n = 2i, Y αn − Y α

n−1 = 2j)

=∑k∈Z

P0(Y αn+1 = 2i+ 2j, Y α

n = 2k, Y αn−1 = 2k − 2j)

=∑k∈Z

P0(Y αn+1 = 2i+ 2k|Y α

n = 2k)P0(Y αn = 2k|Y α

n−1 = 2k − 2j)P0(Y αn−1 = 2k − 2j)

= P0(Y α1 = 2i)P0(Y α

1 = 2j) = P0(Y αn+1 − Y α

n = 2i)P0(Y αn − Y α

n−1 = 2j).

This proves that the random variables Y αn+1 − Y α

n , n ≥ 0, are i.i.d. random variables. The

symmetry is obvious. Analogously we prove that the random variables Y βn+1 − Y β

n , n ≥ 0,are i.i.d. symmetric random variables with respect to the probability measure P1(·). 2

Proposition 5.2.5 If α ∧ β < 1, then the chain Xdp is transient.

Proof. Without loss of generality, let us suppose that α∧β = α < 1. By Proposition 5.2.3,it is enough to prove that the chain Yα is transient. From Proposition 5.2.4 we know thatthe chain Yα is a symmetric random walk on 2Z with respect to the probability measure


P0(·). For every i ∈ Z we have

P0(Y α1 = 2i) = p(0, 2i) +

∑j∈2Z+1

p(0, j)p(j, 2i) + . . . ≥ f(α,c)(2i).

Let ϕ(ξ) be the characteristic function of the distribution P0(Y α1 ∈ ·). From the symmetry

property of the distribution P0(Y α1 ∈ ·), we have

Re

(1

1− ϕ(ξ)

)=

1∑j∈Z

(1− cos(2ξj))P0(Y α1 = 2j)

≤ 1∑j∈Z

(1− cos(2ξj))f(α,c)(2j).

Note that∑

j∈Z cos(2ξj)f(α,c)(2j) is a Fourier transform of a symmetric sub-probabilitymeasure on 2Z. Using completely the same arguments as in Proposition 5.1.3, desire resultfollows directly from from Theorem 2.7.13 (iv). 2

Now, let us consider the general case. Let m ≥ 1, α0, . . . , αm−1 ∈ (0, 2) andc0, . . . , cm−1 ∈ (0,∞) be arbitrary. Let Xdp be a Markov chain on Z given by

α(i) = αj and c(i) = cj

for i ≡ jmod (m), i.e., the functions α : Z −→ (0, 2) and c : Z −→ (0,∞) are periodic func-tions with period m. Furthermore, let us suppose that the probability functions f(αi,ci)(j),i = 0, . . . ,m − 1, satisfy f(αi,ci)(−j) = f(αi,ci)(j) for all j ∈ Z and all i = 0, . . . ,m − 1.Then, it is not hard to prove that Propositions 5.2.1, 5.2.2, 5.2.3 and 5.2.4, except perhapsthe symmetry property of the associated chains (random walks) Yαi , i = 0, . . . ,m− 1, arealso valid in this periodic case. Analogously as in Proposition 5.2.5, by using

Re

(1

1− z

)=

1− a(1− a)2 + b2

≤ 1

1− a

for all z = a+ ib ∈ C such that |z| ≤ 1, we get the following.

Theorem 5.2.6 If α0 ∧ α1 ∧ · · · ∧ αm−1 < 1, then the chain Xdp is transient. 2

Clearly, the above statement should be an if, and only if, statement, i.e., there is no reasonnot to believe that α0 ∧ α1 ∧ · · · ∧ αm−1 = 1 implies the recurrence of the chain Xdp. Butthis case is not covered by Theorem 5.1.1 and it seems to be much more complicated.

As a consequence of Theorem 5.2.6, together with Theorem 5.1.1, we get a weakerversion of Proposition 5.1.3.

Corollary 5.2.7 A random walk with jump probability function f(j), satisfying f(j) =f(−j) for all j ∈ Z, and f(j) ∼ c|j|−α−1, when |j| −→ ∞, where α ∈ (0, 2) and c ∈ (0,∞),is recurrent if α > 1, and it is transient if α < 1. 2


5.3 Recurrence and transience of (α, β)-stable-like Markov

chains

In this section, we derive recurrence and transience conditions for the chain Xd(α,β).The chain Xd(α,β) is a special case of the oscillating random walk. Let µ(·) and ν(·) bedistributions on Z. The oscillating random walk is a Markov chain Y on Z given bythe following transition function

p(i, j) :=

µ(j − i), i < 0

pµ(j) + qν(j), i = 0ν(j − i), i > 0,

where p, q ∈ R+, p+ q = 1. Clearly, under the assumptions

(i) the greatest common divisors of the sets Aµ := i ∈ Z : µ(i) > 0 and Aν := i ∈ Z :ν(i) > 0 are equal to 1

(ii) the set Aµ is unbounded from above

(iii) the set Aν is bounded from below

the oscillating random walk Y is irreducible.

Theorem 5.3.1 [36, Theorem 2] Let us denote by Sµ and Sν the random walks withjump distributions µ(·) and ν(·), respectively. Assume that Sµ is strongly attracted toαµ-stable distribution function Fαµ(x), 0 < αµ ≤ 2, and that −Sν is strongly attractedto αν-stable distribution function Fαν (x), 0 < αν ≤ 2. Define aµ := αµ(1 − Fαµ(0)) andaν := αν(1−Fαν (0)). Then, the oscillating random walk Y is recurrent if aµ + aν > 1, andit is transient if aµ + aν < 1. 2

Proposition 5.3.2 Let α ∈ (0, 2) and c ∈ (0,∞) be arbitrary and let f(α,c) : Z −→ R bean arbitrary probability function such that f(α,c)(j) ∼ c|j|−α−1, when |j| −→ ∞. Let usassume that f(α,c)(−j) = f(α,c)(j) holds for all j ∈ Z if α = 1 and

∑j∈Z jf(α,c)(j) = 0 holds

if α > 1. Then, the random walk S with jump distribution(. . . − 1 0 1 . . .. . . f(α,c)(−1) f(α,c)(0) f(α,c)(1) . . .

)is strongly attracted to a SαS distribution.

Proof. Let F (x) =∑

i≤x f(α,c)(i) be the distribution function of the random walk S. ByTheorem 2.2.5, it suffices to show that there exist constants c1, c2 ∈ R+, c1 + c2 > 0, suchthat

F (−x)

1− F (x)−→ c1

c2

and1− F (x) + F (−x)

1− F (kx) + F (−kx)−→ kα


for all k > 0, as x −→ ∞. Let 0 < ε < 1 be arbitrary. Then, by assumption, there existsi0 ∈ N such that for all |i| ≥ i0 we have∣∣∣∣f(α,c)(i)

|i|α+1

c− 1

∣∣∣∣ < ε. (5.5)

For x > 0 such that dxe ≥ i0, by applying (5.5), we have

c(1− ε)∑∞

i=dxe1

iα+1

c(1 + ε)∑∞

i=bx+1c1

iα+1

<F (−x)

1− F (x)=

∑∞i=dxe f(α,c)(−i)∑∞i=bx+1c f(α,c)(i)

<c(1 + ε)

∑∞i=dxe

1iα+1

c(1− ε)∑∞

i=bx+1c1

iα+1

.

Since for every i > 1

1

α(i+ 1)α=

∫ ∞i+1

dy

yα+1<∞∑j=i

1

jα+1<

∫ ∞i−1

dt

yα+1=

1

α(i− 1)α,

we have(1− ε)(bx+ 1c − 1)α

(1 + ε)(dxe+ 1)α<

F (−x)

1− F (x)<

(1 + ε)(bx+ 1c+ 1)α

(1− ε)(dxe − 1)α.

Now, by letting x −→∞ and ε −→ 0 we get

limx−→∞

F (−x)

1− F (x)= 1.

Hence, we can take arbitrary c1 = c2 > 0. Analogously,

limx−→∞

1− F (x) + F (−x)

1− F (kx) + F (−kx)= kα

for all k > 0. Thus, we have proved that the random walk S is attracted to an Sα(β, γ, δ)distribution. Finally, the strong attraction follows form (2.13). 2

Theorem 5.3.3 If the probability functions f(α,c)(j) := fi(j), for i < 0, and f(β,d)(j) :=

fi(j), for i ≥ 0, appearing in the definition of the chain Xd(α,β), satisfy f(α,c)(j) = f(α,c)(−j)and f(β,d)(j) = f(β,d)(−j) for all j ∈ Z, then the chain Xd(α,β) is recurrent if α+β > 2, andit is transient if α + β < 2.

Proof. According to Theorem 5.3.1 and Proposition 5.3.2, we only have to show that thelimiting stable distribution functions Fα(x) and Fβ(x) satisfy Fα(0) = Fβ(0) = 1

2. But,

this follows from the symmetry assumption of probability functions f(α,c)(j) and f(β,d)(j)and Levy’s continuity theorem. Moreover, limiting stable distributions are symmetric dis-tributions. 2

Clearly, as in the general state space, α + β = 2 should imply recurrence of the chainXd(α,β). But this case is not covered by Theorem 5.3.1 and it seems to be much morecomplicated. As a consequence of Theorem 5.3.3 we get the following corollary.


Corollary 5.3.4 A random walk with jump probability function f(j), satisfying f(j) =f(−j) for all j ∈ Z, and f(j) ∼ c|j|−α−1, when |j| −→ ∞, where α ∈ (0, 2) and c ∈ (0,∞),is recurrent if α > 1, and it is transient if α < 1. 2

Bibliography

[1] M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions withformulas, graphs, and mathematical tables. John Wiley & Sons Inc., New York, 1984.

[2] R. F. Bass. Uniqueness in law for pure jump Markov processes. Probab. Theory RelatedFields, 79(2):271–287, 1988.

[3] A. Bensoussan, J-L. Lions, and G. Papanicolaou. Asymptotic analysis for periodicstructures, volume 5. North-Holland Publishing Co., Amsterdam, 1978.

[4] P. Billingsley. Convergence of probability measures. John Wiley & Sons Inc., NewYork, second edition, 1999.

[5] R. M. Blumenthal and R. K. Getoor. Markov processes and potential theory. AcademicPress, New York, 1968.

[6] R. M. Blumenthal, R. K. Getoor, and D. B. Ray. On the distribution of first hits forthe symmetric stable processes. Trans. Amer. Math. Soc., 99:540–554, 1961.

[7] B. Bottcher. An overshoot approach to recurrence and ransience of Markov processes.Stochastic Process. Appl., 121(9):1962–1981, 2011.

[8] B. Bottcher and R. L. Schilling. Approximation of Feller processes by Markov chainswith Levy increments. Stoch. Dyn., 9(1):71–80, 2009.

[9] L. Breiman. Probability, volume 7. Society for Industrial and Applied Mathematics(SIAM), Philadelphia, PA, 1992.

[10] K. L. Chung. A course in probability theory. Academic Press Inc., San Diego, CA,third edition, 2001.

[11] P. Courrege. Sur la forme integro-differentielle des operateus de C∞K dans C satisfaisantau principe du maximum. Sem. Theorie du Potentiel, expose 2:38 pp., 1965-1966.

[12] R. Durrett. Probability: theory and examples. Cambridge University Press, Cambridge,fourth edition, 2010.

[13] S. N. Ethier and T. G. Kurtz. Markov processes. John Wiley & Sons Inc., New York,1986.

107

108 BIBLIOGRAPHY

[14] W. Feller. An introduction to probability theory and its applications. Vol. II. JohnWiley & Sons Inc., New York, second edition, 1971.

[15] B. Franke. The scaling limit behaviour of periodic stable-like processes. Bernoulli,12(3):551–570, 2006.

[16] B. Franke. Correction to: “The scaling limit behaviour of periodic stable-like pro-cesses” [Bernoulli 12 (2006), no. 3, 551–570]. Bernoulli, 13(2):600, 2007.

[17] W. Gawronski. On the bell-shape of stable densities. Ann. Probab., 12(1):230–242,1984.

[18] B. V. Gnedenko and A. N. Kolmogorov. Limit distributions for sums of indepen-dent random variables. Addison-Wesley Publishing Company, Inc., Cambridge, Mass.,1954.

[19] N. Jacob. Pseudo differential operators and Markov processes. Vol. I. Imperial CollegePress, London, 2001.

[20] N. Jacob. Pseudo differential operators & Markov processes. Vol. II. Imperial CollegePress, London, 2002.

[21] N. Jacob. Pseudo differential operators and Markov processes. Vol. III. ImperialCollege Press, London, 2005.

[22] J. Jacod and A. N. Shiryaev. Limit theorems for stochastic processes, volume 288.Springer-Verlag, Berlin, second edition, 2003.

[23] J. H. B. Kemperman. The oscillating random walk. Stochastic Process. Appl., 2:1–29,1974.

[24] V. N. Kolokoltsov. Symmetric stable laws and stable-like jump-diffusions. Proc.London Math. Soc. (3), 80(3):725–768, 2000.

[25] V. N. Kolokoltsov. Markov processes, semigroups and generators, volume 38. Walterde Gruyter & Co., Berlin, 2011.

[26] J. Lamperti. Criteria for the recurrence or transience of stochastic process. I. J. Math.Anal. Appl., 1:314–330, 1960.

[27] M. V. Menshikov, I. M. Asymont, and R. Yasnogorodskij. Markov processes withasymptotically zero drift. Probl. Inf. Transm., 31(3):60–75, 1995.

[28] S. P. Meyn and R. L. Tweedie. Generalized resolvents and Harris recurrence of Markovprocesses. In Doeblin and modern probability (Blaubeuren, 1991), volume 149, pages227–250. Amer. Math. Soc., Providence, RI, 1993.

[29] S. P. Meyn and R. L. Tweedie. Markov chains and stochastic stability. Springer-VerlagLondon Ltd., London, 1993.

BIBLIOGRAPHY 109

[30] S. P. Meyn and R. L. Tweedie. Stability of Markovian processes. II. Continuous-timeprocesses and sampled chains. Adv. in Appl. Probab., 25(3):487–517, 1993.

[31] A. Negoro. Stable-like processes: construction of the transition density and the be-havior of sample paths near t = 0. Osaka J. Math., 31(1):189–214, 1994.

[32] E. Nummelin. General irreducible Markov chains and nonnegative operators, vol-ume 83. Cambridge University Press, Cambridge, 1984.

[33] P. E. Protter. Stochastic integration and differential equations, volume 21. Springer-Verlag, Berlin, second edition, 2004.

[34] D. Revuz. Markov chains, volume 11. North-Holland Publishing Co., Amsterdam,second edition, 1984.

[35] D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293.Springer-Verlag, Berlin, third edition, 1999.

[36] B. A. Rogozin and S. G. Foss. The recurrence of an oscillating random walk. Teor.Veroyatn. Primen., 23(1):161–169, 1978.

[37] G. Samorodnitsky and M. S. Taqqu. Stable non-Gaussian random processes. Chapman& Hall, New York, 1994.

[38] N. Sandric. Recurrence and transience property for a class of Markov chains. Preprint,2012.

[39] N. Sandric. Recurrence and transience property for two cases of stable-like Markovchains. Preprint, 2012.

[40] K-I. Sato. Levy processes and infinitely divisible distributions, volume 68. CambridgeUniversity Press, Cambridge, 1999.

[41] R. L. Schilling. Conservativeness and extensions of feller semigroups. Positivity,2:239–256, 1998.

[42] R. L. Schilling. Growth and Holder conditions for the sample paths of Feller processes.Probab. Theory Related Fields, 112(4):565–611, 1998.

[43] R. L. Schilling and J. Wang. Some theorems on Feller processes: transience, localtimes and ultracontractivity. To appear in: Trans. Amer. Math. Soc., 2012.

[44] J. A. Schnurr. The Symbol of a Markov Semimartingale. PhD thesis, der FakultatMathematik und Naturwissenschaften der Technischen Universitat Dresden, 2009.

[45] F. Spitzer. Principles of random walks. Springer-Verlag, New York, second edition,1976.

[46] R. L. Tweedie. Topological conditions enabling use of Harris methods in discrete andcontinuous time. Acta Appl. Math., 34(1-2):175–188, 1994.

110 BIBLIOGRAPHY

[47] V. M. Zolotarev. One-dimensional stable distributions, volume 65. American Mathe-matical Society, Providence, RI, 1986.

BIBLIOGRAPHY 111

112 BIBLIOGRAPHY

Acknowledgements

I would like to thank my supervisor Prof. Zoran Vondracek for the patient guidance andsupport.I am grateful to my parents for believing in me.Special thanks to Marina, for understanding and constant and unconditional support.

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