Second-order properties of thresholded realized power variations of FJA

additive processes

Jose E. Figueroa-Lopez∗ Jeffrey Nisen†

March 30, 2019

Abstract: For a class of additive processes of finite jump activity (FJA), we give precise conditions for the mean-squared

consistency and feasible Central Limit Theorems of thresholded realized power variation estimators (TRV). To justify that the

proposed conditions are the “best possible”, we also show that these are necessary for FJA Levy processes. Non-asymptotic

upper bounds and asymptotic decompositions of the mean-squared errors of our estimators are also provided. For comparison

purposes, we also obtain the analogous asymptotic decomposition for a general multi-power realized variation (MPV). These

results theoretically justify the relatively large bias of MPV (when compared to TRV) observed numerically in earlier Monte

Carlo studies.

Keywords: Truncated realized variations, multipower realized variations, integrated variance estimation, jump features esti-

mation, Levy processes, additive processes, nonparametric estimation.

1 Introduction

Together with multi-power realized variations (MPVs), thresholded realized power variations (TPV) are the most

popular estimation devices to synthesize the continuous volatility component of an Ito semimartingale X := (Xt)t≥0

in the presence of a jump component. The asymptotic properties of TPVs are relatively well understood with an

emphasis to establish central limit theorems (CLTs). By now, there are simple sufficient conditions for the consistency

and CLTs of TPVs. In the particular case of a thresholded realized quadratic variation (TQV), Mancini (2009), who

first introduced the idea of thresholding in Mancini (2001, 2003), established consistency and a CLT for a quite general

class of Ito semimartingales1, under the assumption that the threshold sequence, hereafter denoted by Bn, is such that

limn→0

Bn√hn ln (1/hn)

=∞, (1.1)

∗Department of Mathematics and Statistics, Washington University in St. Louis, MO, 63130, USA (figueroa-lopez@wustl.edu).†Quantitative Analytics, Barclays, New York, NY, 10019, USA (jeffrey.nisen@barclayscapital.com).1Though, the CLT was only established for finite jump activity processes.

1

where hn denotes the time span between consecutive sampling observations. Roughly, the reference threshold Bn :=√hn ln(1/hn) is related to the Brownian motion’s modulus of continuity so that if an increment of the process exceeds

Bn, in absolute value, and hn is small enough, it is almost certain that such a “large” increment was due to the presence

of a jump rather than a “typical” increment of the diffusive component. However, the condition (1.1) excludes some

useful threshold sequences proposed in the literature such as in Andersen et al. (2007a), Gegler and Stadtmuller

(2010), Figueroa-Lopez and Nisen (2013), and Figueroa-Lopez and Mancini (2018) (see below for more details). Jacod

(2008) (in addition to Jacod (2007)) is also an important reference where, in a quite general semimartingale setting,

consistency and CLTs are obtained for certain TPVs. In particular, only power type thresholds of the form Bn = αhωn

are analyzed therein. See also the monograph Jacod and Protter (2012) for an in-depth review of these and other new

results.

In spite of the literature surveyed in the previous paragraph, it comes as a surprise that a precise study of the second-

order properties (bias and variance) of these estimators is absent in the literature. In this work, we determine sufficient

conditions for the mean-squared-error (MSE) consistency of threshold-based estimators of not only the continuous

component but also some jumps features. Non-asymptotic upper bounds for the MSE of the different estimators are

also considered. This is done within a certain class of Finite-Jump-Activity (FJA) additive processes; i.e., processes

of independent increments with finitely many jumps in any finite time horizon. Our results can be extended to cover

processes that are conditionally additive, given a suitable filtration. In particular, this includes the case where the drift,

volatility, and intensity of jumps can be stochastic, but independent of the Brownian motion and Poisson processes

driving the state process X. The setting studied here is more restrictive than the general semimartingale setting usually

considered in the literature of nonparametric high-frequency methods, which are inspired by financial applications (e.g.,

as in Jacod and Protter (2012)). Nevertheless, finite jump activity models are still widely used in many applications

and areas (including finance) because of their easy interpretability and tractability. Certain nonparametric tests (such

as those related to the Blumenthal-Getoor index) would need to be carried out beforehand in order to justify the

applicability of our results. See also the conclusion section for further information regarding possible extensions of our

results to more general settings.

In order to justify that the proposed upper bounds and thereof sufficient conditions are the “best” one can hope for,

we show that the given conditions are necessary when working with FJA Levy models, in which case, we furthermore

provide precise asymptotic decompositions of the bias and variance of the estimators that are of the same order as the

derived upper bounds. The latter decompositions in turn help to explain how the different model parameters affect the

performance of the estimators and the best possible rates of convergence. In addition, these decompositions provide a

theoretical justification as to why TPVs tend to exhibit smaller bias when numerically compared to MPVs, a fact that

has previously been documented in the literature (see, e.g., Corsi et al. (2010)). To that end, we also derive precise

asymptotics for the second-order properties of MPVs. Another application of the asymptotic expansions considered

in this work is in finding criteria for tuning the threshold parameter. Concretely, a natural idea is to consider the

threshold that minimizes the leading order terms of the Mean Square Error (MSE). Following this idea, we show that

2

this threshold parameter is the same (asymptotically) as the threshold that minimizes the MSE and that was derived

in Figueroa-Lopez and Mancini (2018).

We now proceed to briefly preview our results for a finite time horizon T . Throughout, Bn represents the threshold

used when the time span between observations is hn and we assume that Bn → 0 and hn → 0, as n → ∞. As

mentioned before, the central question is how fast the threshold Bn should converge to 0 for the MSE consistency of

the estimators of different process features. In the FJA additive framework described above, a sufficient condition for

the MSE consistency of the TQV is simply given by

limn→∞

Bn√hn

=∞, (1.2)

which also turns out to be necessary for a FJA Levy process. Condition (1.2) puts an “upper bound” on the rate at

which Bn can converge to 0, i.e., a maximal speed (Bn cannot go to 0 faster than√hn). Interestingly enough, this is

also a sufficient condition for the MSE consistent estimation of the jump component, say JT , of the process of X via a

threshold type estimator as introduced in Mancini (2004) (see (3.12) for the details). The reason of this equivalence lies

in the fact that both estimators are “highly” affected by “false positives” jump detections (i.e., incorrectly indicating

that a jump is present in an increment), while being far less sensitive to “false negative” jump detections (i.e., missing

the presence of a jump in an increment). Condition (1.2) is designed to provide the right control of both types of

errors.

In the case of estimating the total number of jumps NT during a fixed time horizon [0, T ], a sufficient condition for

the MSE consistency of the threshold-based estimator NT = #i : |∆ni X| > Bn is given by

limn→0

1

Bn√hnφ

(Bn

σT√hn

)= 0, (1.3)

where σT is an upper bound on the volatility of the process over [0, T ] and φ(·) denotes the standard Gaussian

probability density function. The condition (1.3) is proved to also be necessary for FJA Levy process. This result

says that, when it comes to recovering the total number of jumps, we need more stringent conditions than (1.2). This

is expected since, by design, the estimator of NT is highly sensitive to both false positive and false negative jump

detections and the condition (1.3) is designed to give a stricter control of both types of errors. It is relevant pointing

out here that if one had no idea of an upper bound for the volatility during the specified time horizon (i.e., if we wished

that the condition (1.3) is satisfied regardless of how high the value of σT may be), then the sequence (Bn)n≥1 would

necessarily have to satisfy (1.1). However, in financial applications, it is often the case that one has a good sense of

what are reasonable volatility levels and, therefore, one can take advantage of this information to design a suitable

threshold estimator.

We also consider conditions for the validity of a feasible CLT. In that case, the conditions

limn→∞

Bn√hn

=∞, limn→0

Bnhn

φ

(Bn

σT√hn

)= 0, (1.4)

are sufficient. The latter is weaker than (1.1) and now covers threshold sequences of the form Ban :=√

2σ2hn log(1/hn),

for σ ≥ σT . Sequences of the form Ban were studied in Figueroa-Lopez and Nisen (2013) and Figueroa-Lopez and

3

Mancini (2018) as a byproduct of well-posed optimal thresholding problems. Other sequences proposed in the literature

that comply with (1.4), but not with (1.1), are of the form

Bn[BF (σ, C)] = σh1/2n Φ−1

(1− C

2hn

), (1.5)

which was independently proposed by Andersen et al. (2007a) and Gegler and Stadtmuller (2010) and is inspired by

a Bonferroni Type I error control procedure commonly used in applied statistics for large-scale hypothesis testing.

Our results show that, even though a strong condition such as (1.1) seems to be needed for an asymptotically perfect

classification of jumps (see the first assertion in Theorem 4.3 below), this is not necessary for a consistent estimation

and a feasible CLT of the estimator, and excludes some threshold sequences proposed in the literature. One could argue,

as it is often done in the literature, that the differences between (1.1) and (1.4) are merely logarithmic terms, which

are negligible. However, these differences may have noticeable effects in realistic finite sample settings as shown in the

extensive simulation experiments of Figueroa-Lopez and Mancini (2018). For instance, one of the power thresholds

that, to our knowledge, performs the best is

BJT = 4h0.49 ˆσBP ,

where ˆσ2BP is an estimate of σ2 :=

∫ T0σ2sds/T based on the standard Bipower variation of Barndorff-Nielsen and

Shephard (2004). The threshold above was used in Jacod and Todorov (2015). In Figueroa-Lopez and Mancini

(2018), the performance of BJT was compared to certain data-driven thresholds proportional to√h log(1/h). For a

Heston model with Merton type jumps and 1-month 5 minute data, the latter thresholds (asymptotically equivalent

to√h log(1/h)) could attain between 40 to 80% MSE reduction. In our opinion, this reduction is not negligible.

The rest of the paper is organized as follows. Section 2 introduces the framework and some notation commonly

used throughout the manuscript. An introduction to the problem and thresholded power variation estimators is given

in Section 3. The main results are presented in Section 4. Section 5 contains some concluding remarks. For a

more streamlined presentation, the proofs of the main results and other needed lemmas are given in three separate

appendices.

2 Framework and Notation

In this section we introduce the model framework and standing assumptions. Throughout, we let W = (Wt)t≥0 and

N = (Nt)t≥0 respectively denote a Wiener process and a non-homogeneous Poisson process with deterministic intensity

(λt)t≥0 on a complete filtered probability space (Ω,F , (Ft)t≥0,P). Furthermore, given deterministic functions γ and

σ and an absolutely continuous probability distribution F on (R,B(R)) with density f(·), we consider a Finite-Jump

Activity (FJA) additive process X = (Xt)t≥0 of the form

Xt := Xct + Jt :=

(∫ t

0

γsds+

∫ t

0

σsdWs

)+

Nt∑i=1

ζi, (2.1)

4

where ζjj≥1 denote the successive jumps of the jump component J , which are by definition i.i.d. with distribution

F . Hereafter, we denote the arrival time of the jth jump ζj by τj . Throughout, we assume that

γT := sups≤T|γs| <∞, λT := sup

s≤Tλs <∞, σT := sup

s≤Tσs <∞, σT := inf

s≤Tσs > 0, (2.2)

for any fixed T ∈ (0,∞). We further require that t 7→ σ2t is continuous. In the case where γ, σ, and λ are constant

functions, the process X becomes a Levy process and will be referred to as a FJA Levy model.

The restrictions of the jump density f on both (0,∞) and (−∞, 0) are assumed to be bounded such that f(0+), f(0−) ∈

(0,∞). More specifically, we assume that f can be written as

f(x) = pf+(x)1[x≥0] + qf−(x)1[x<0], (2.3)

where p ∈ [0, 1], q := 1 − p, and f+ : [0,∞) → [0,∞) and f− : (−∞, 0] → [0,∞) are bounded density functions such

that

f±(0) = limx→0±

f±(x) ∈ (0,∞). (2.4)

Throughout,

C0(f) := limε→0+

1

2ε

∫ ε

−εf(x)dx =

1

2[pf+(0) + qf−(0)] , C(f) := pf+(0)− qf−(0). (2.5)

The constant C0(f) quantifies the mass concentration of f around the origin. Note that C0(f) = f(0) and C(f) = 0 if

f is continuous at the origin.

For estimation purposes, we assume that, for each n ∈ N, our data consists of Mn regular observations from the

process X over a finite time horizon [0, Tn]. The time span between observations as well as the resulting sampling grid

are respectively denoted by hn := Tn/Mn and ti := tni := ihn, for i = 1, . . . ,Mn. As usual,

∆ni Y := Yti − Yti−1

,

denotes the ith increment of a generic process (Yt)t≥0. In what follows, we consider two asymptotic regimes:

i) T := limn→∞

Tn ∈ (0,∞), limn→∞

hn = 0, or ii) T := limn→∞

Tn =∞, limn→∞

hn = 0

The first (respectively, second) setup is referred to as a high-frequency/finite-horizon (respectively, high-frequency/long-

horizon) sampling scheme.

Let us finish this section with some recurrent notation used throughout the paper:

• Φ(·) and φ(·) denote the standard Gaussian distribution and density functions, respectively, while Φ−1(y) :=

infx : Φ(x) ≥ y and Φ(x) =∫∞xφ(z)dz respectively denote the corresponding quantile and survival functions.

The below well-known Gaussian tail estimates will play key roles in some proofs:

Φ(x) ≤ e−x2/2

x√

2π, (x > 0), and Φ(x) ∼ e−x

2/2

x√

2π, (x→∞). (2.6)

5

It is also useful to recall the following well-known properties of a standard normal variable Z for any k > 0 and

x ∈ R:

mk := E[|Z|k

]=

2k2

√π

Γ

(k + 1

2

), E [Z1Z>x] = φ(x), E

[Z21Z>x

]= xφ(x) + Φ(x). (2.7)

• For brevity we will often use the constants

γn,i :=

∫ ti

ti−1

γsds, σ2n,i :=

∫ ti

ti−1

σ2sds, λn,i =

∫ ti

ti−1

λsds, (2.8)

and, in the case of a FJA Levy model, we often drop the subindex i.

• Given two sequences (an)n≥1 and (bn)n≥1 of positive reals, we write an = o(bn) or an bn (resp., an = O(bn)

or an bn) if limn→∞ an/bn = 0 (resp., lim supn→∞ an/bn <∞).

• Given ` vanishing sequences a(1)n → 0, . . . , a

(`)n → 0, the terminology

An = a(1)n + · · ·+ a(`)

n + h.o.t.,

means that An = a(1)n + · · · + a

(`)n + Rn, where Rn = o(a

(1)n ) + · · · + o(a

(`)n ), as n → ∞. Here, h.o.t. refers to

“higher order terms”2.

• Given an estimator θ of a parameter θ, we set

Bias(θ; θ)

:= E[θ]− θ, MSE

(θ; θ)

:= E[(θ − θ

)2]

= Var(θ)

+ Bias(θ; θ)2

.

3 Multipower and Thresholded Power Variation Estimators

In this section we introduce some of the most commonly used nonparametric estimators for the integrated volatility of

a general Ito semimartingale, whose prototypical form is given by

Xt := x+

∫ t

0

γsds+

∫ t

0

σsdWs + Jt,

where J is a pure-jump process. In the literature, one finds two broad classes: multipower and threshold type estimators.

We now proceed to give their definitions and a heuristic description of their respective mechanics.

3.1 Multi-Power Variation Estimators

The precursor of the general multi-power variation (MPV) estimators was the Bi-Power Variation Estimator (BPV),

originally introduced by Barndorff-Nielsen and Shephard (2004). Given the consecutive increments (∆ni X)Mn

i=1 of a

process X, the BPV estimator is defined as

BPV (X)nt :=π

2

∑i:tni+1≤t

|∆ni X||∆n

i+1X|, t ≤ T. (3.1)

2It is worth mentioning that in some literature a “higher order term” is meant to denote a term of the form o(a(1)n + · · ·+ a

(`)n ). This is

not the convention in this paper.

6

In turn, the latter estimator is inspired by the traditional realized quadratic variation,

QV (X)nt :=∑i:tni ≤t

|∆ni X|2, (3.2)

which converges to the quadratic variation of a general semimartingale X; i.e.,

P− limn→∞

n∑i=1

(Xsni−Xsni−1

)2

=

∫ t

0

σ2sds+

∑0≤s≤t

(4Xs)2 =: [X,X]t, (3.3)

where (sni )ni=1 is any sequence of stopping times such that 0 = sn0 < sn1 < · · · < snn = t and max1≤i≤n(sni − sni−1) → 0

a.s. as n→∞.

For a continuous process, it is clear from (3.3) that the integrated volatility∫ t

0σ2sds can be recovered via the

realized variation. However, in the presence of jumps, the latter assertion does not hold anymore, while (3.1) is still

able to recover the integrated variance. Heuristically, this follows since |∆ni X||∆n

i+1X| ≈ 0 whenever there is at most

one “big” jump in [ti−1, ti+1], as it is unlikely that two such jumps would occur in two adjacent subintervals, while

|∆ni X||∆n

i+1X| ≈ (∆ni X)2, otherwise. The constant π/2 in (3.1), which coincides with E[|Z|]−2 for a standard Gaussian

variable Z, is necessary for the consistency of the estimator to hold.

As previously mentioned BPV estimators were subsequently extended to Multi-Power Variation Estimator (MPV)

by Barndorff-Nielsen and Shephard in a series of papers (see Barndorff-Nielsen and Shephard (2006) and references

therein). For given positive reals r = (r1, . . . , rk) ∈ Rk+, the corresponding MPV estimator is defined by

MPV (X)n[r]t := h1−r+/2

n C(r)−1∑

i:tni+k−1≤t

k∏j=1

|∆i+j−1X|rj , t ≤ T, (3.4)

where r+ := r1 +· · ·+rk and C(r) :=∏ki=1 E[|Z|ri ] for Z =D N(0, 1). The estimator (3.1) is recovered with r1 = r2 = 1

and, again, the constant C(r) in (3.4) is chosen so that∫ t

0σr

+

s ds is consistently estimated.

Two appealing features of multi-power variation style estimators are: 1) they are relatively easy to use in practice

since they require minimal parameter tune-up, and 2) they have somewhat well understood in-fill asymptotic properties

(see e.g. Barndorff-Nielsen and Shephard (2004), Woerner (2006), Barndorff-Nielsen et al. (2006), Jacod (2008), and

Vetter (2010)). Although, by design, they are incapable of detecting jumps in the underlying process, they can be used

in combination with the realized quadratic variation estimator (3.2) to estimate the jump component of the quadratic

variation (see Veraart (2010) for further asymptotic results in this regard). Additionally, differences or ratios of power

variations estimators have also been used in the development of several non-parametric tests for the presence of a jump

component. We refer the interested reader to Barndorff-Nielsen and Shephard (2006), Jiang and Oomen (2008), Lee

and Mykland (2008), Aıt-Sahalia and Jacod (2009b), and Fan and Fan (2011) for more details regarding non-parametric

tests for the presence of a jump component.

Despite their appealing properties, lower order multi-power variations estimators, such as the bi-power or tri-power

variations, exhibit comparatively high biases in the presence of jumps, even for reasonably large sample sizes. The

7

interested reader is directed to Barndorff-Nielsen and Shephard (2004), Huang and Tauchen (2005), Andersen et al.

(2007a), and Corsi et al. (2010) for more information about the finite sample behavior of the BPV estimator. The

inherent lack of robustness associated with these estimators prompted some modified estimators based on the rank

statistics of adjacent absolute increments of the process (see, e.g., Andersen et al. (2007b) and Christensen et al.

(2010)). Threshold estimation gives an alternative to multipower-based estimators.

3.2 Threshold Estimators

Throughout the rest of the paper, a thresholding sequence B = (Bn)n≥1 is a deterministic sequence of positive real

numbers such that

limn→∞

Bn = 0. (3.5)

The class of thresholded power variations began with the work of Mancini (2001, 2003). Given a thresholding sequence

B and a positive integer k, the kth-order Thresholded Realized Power Variation (TPV) estimator is defined by

TPV (X)[B]n,kt :=1

mkhk/2−1n

∑i:tni ≤t

|∆ni X|

k1|∆n

i X|≤Bn, t ≤ T. (3.6)

The special case of k = 2 is commonly referred to as the Thresholded Realized Quadratic Variation (TQV) estimator:

TQV (X)[B]nt :=∑i:tni ≤t

|∆ni X|21|∆n

i X|≤Bn. (3.7)

Corsi et al. (2010) subsequently combined both the thresholding and multi-power variation approaches and proposed

the so-called Thresholded Bi-Power and Multi-Power Variation estimators, which are defined as follows:

TBPV (X)[B]nt :=π

2

∑i:tni+1≤t

|∆ni X||∆n

i+1X|1|∆ni X|≤Bn1|∆n

i+1X|≤Bn, (3.8)

TMPV (X)[B]n[r]t := h1−r+/2

n C(r)−1∑

j:tnk+j−1≤t

k+j−1∏i=j

|∆ni X|ri−j+11|∆n

i X|≤Bn. (3.9)

The basic idea of thresholding estimators is to filter out those increments which, due to their “large” magnitude when

compared to a suitably chosen threshold parameter, may contain “big” jumps. Several explicit functional forms for

thresholding sequences have been proposed in the literature. A power threshold of the form

Bn[Pow(α, ω)] := αhωn , (3.10)

for some constant α > 0 and ω ∈ (0, 1/2), is the most popular in the literature. Both Andersen et al. (2007a) and

Gegler and Stadtmuller (2010) independently proposed the threshold

Bn[BF (σ, C)] = σh1/2n Φ−1

(1− C

2hn

), (3.11)

8

for some constant C and some prior estimate σ of σ. Recall that Φ−1 denotes the quantile function of a standard

normal variable. The resulting TPV estimator is inspired by the Bonferroni Type I error control procedure commonly

used in applied statistics for large-scale hypothesis testing.

Several important in-fill asymptotic properties such as consistency and central limit theorems are known under

somewhat general conditions on the threshold sequence (see, e.g., Mancini (2004, 2009) and Jacod (2008)). As their

unthresholded counterparts, these estimators can also be used to construct non-parametric tests for some path-wise

properties of the underlying process. For example, under a model assumption of a Brownian semi-martingale plus a

pure-jump Levy component of bounded variation, Cont and Mancini (2011) used realized power variation estimators to

build tests for the presence of a continuous martingale component. See also Aıt-Sahalia and Jacod (2009a) for further

applications of the TPV statistics in estimating the degree of jump activity in general semimartingale models.

TPV estimators can also be applied in a natural way for consistently estimating the jump features of the process

in a high-frequency/long-horizon sampling scheme. The basic idea is to use the increment ∆ni X = Xti − Xti−1

as a

proxy of a jump that has likely occurred during the interval [ti−1, ti] based on the employed threshold criterion. The

following statistics, already studied in Mancini (2003), are then natural estimates for the underlying compound Poisson

process (Jt)t≥0 of the jump-diffusion (2.1) and its corresponding jump counting process (Nt)t≥0, respectively:

J [B]nt :=∑i:tni ≤t

(∆ni X)1|∆n

i X|>Bn, N [B]nt :=∑i:tni ≤t

1|∆ni X|>Bn. (3.12)

The above estimators leads to natural estimators of the successive times and sizes (τj , ζj)j≥1 of the jumps of J as

τnj := min ti = ihn : ti > τj−1 & |∆ni X| > Bn , ζnj := ∆n

τj/hnX, (j ≥ 1), (3.13)

where τ0 := 0 and min ∅ := ∞. In a finite time horizon sampling framework (i.e., limn→∞ Tn = T < ∞), Mancini

(2009)’s Theorem 1 implies the strong consistency of the previous estimators to their corresponding true values:

J [B]nTna.s.−→ JT , N [B]nTn

a.s.−→ NT , τnj 1τnj ≤Ta.s.−→ τj 1τj≤T , ζnj 1τnj ≤T

a.s.−→ ζj 1τj≤T ,

as n→∞, for each fixed T > 0.

Although the estimators given in (3.6) and (3.7) have many useful attributes, the choice of the thresholding sequence

significantly affects the finite-sample performance of the resulting estimates. As one would expect, not all sequences

(Bn)n≥0 of positive numbers tending to zero will yield useful threshold type estimators. Intuitively, in order for a

threshold estimator to perform well, the threshold sequence should not converge to zero too fast, otherwise there would

be an over filtering of increments that do not actually contain jumps. Conversely, if the rate of convergence is too slow,

then the estimator will include many increments that contain jumps, hence exhibiting poor performance. It is then

apparent that there is a trade-off between speed and accuracy for which a properly chosen threshold sequence should

balance both attributes. In what follows, we shall identify different classes of threshold sequences based on the speed

at which they converge to zero. As it will turn out, the performance of the corresponding threshold estimators in each

class will exhibit qualitatively different properties. The comparison is made in terms of the rates of convergence of

9

their bias, variance, and mean-squared error (MSE) for estimators of both the jump and continuous components. We

also compare their performance with those of multipower variation estimators.

4 Asymptotic Properties of the Estimators

In this section we state necessary and sufficient conditions for threshold type estimators to consistently recover various

model parameters and components under a mean-squared loss function. It is expected that the statistical performance

of a threshold type estimator is connected by a common thread, namely their inherent ability to avoid spurious jump

detection. For any given sampled increment ∆ni X, two principal types of jump detection errors can occur: false positive

or false negative misclassifications. Due to their close connection with statistical hypothesis testing and comparative

relevance with each other, we will refer to these mistakes as Type I and Type II errors, respectively. Concretely,

given i ∈ 1, . . . ,Mn and a threshold sequence (Bn)n, we say that a Type I (resp., Type II) jump misclassification

error has occurred during the time interval [(i−1)hn, ihn] if the indicator 1Type I Errori := 1|∆ni X|>Bn,∆n

i N=0 (resp.,

1Type II Errori := 1|∆ni X|≤Bn,∆n

i N 6=0) is one.

4.1 Continuous Component Estimators

In this section we examine and compare the rates of convergence of the bias, variance, and mean-squared error for the

Thresholded Realized Power Variation (TPV) and the Multipower Variation (MPV) estimators introduced in Section

3.1. Throughout, we only consider a finite-horizon sampling scheme and, thus, for ease of notation, we fix Tn ≡ T . In

what follows, we make use of the terminology h.o.t. (higher order terms) introduced at the end of Section 2. All the

proofs in this section are presented in Appendix A.

We first consider the TPV estimators defined in (3.6).

Theorem 4.1. Suppose the jump distribution has a bounded density of the form (2.3) and B = (Bn)n is a given

thresholding sequence. Then, for an even positive integer k, the kth order thresholded realized variation (3.6) will

converge in mean-squared error to∫ T

0σkudu provided that the following two conditions are satisfied:

(i) limn→∞

Bn√hn

=∞, (ii) limn→∞

B2k+1n

hk−2n

= 0. (4.1)

Furthermore, for a FJA Levy model and an arbitrary positive integer k, the conditions in (4.1) are also necessary and,

in that case, the rates of convergence of the bias and variance, as n→∞, are given by:

E[TPV (X)[B]n,kT

]− Tσk = −2Tσ

mkφ

(Bn

σh1/2n

)(Bn

h1/2n

)k−1

+ TΛ(1)n hn +

2TλC0(f)

(k + 1)mk

Bk+1n

hk/2−1n

+ h.o.t., (4.2)

Var(TPV (X)[B]n,kT ) =2TλC0(f)

m2k(2k + 1)

B2k+1n

hk−2n

+ TΛ(2)n hn + h.o.t., (4.3)

10

where Λ(1)n and Λ(2) are constants given by

Λ(1)n :=

σk−2

2Γ(k+1

2

) [γ2

(k

2

)Γ

(k − 1

2

)− 2λσ2Γ

(k + 1

2

)]− 1[k=1]

γ2(1 + 21[γ<0])

mk

√2πσ2

,

Λ(2)n :=

1

m2k

σ2k2k

π1/2Γ

(2k + 1

2

)− 1

m2k

σ2k2k

πΓ2

(k + 1

2

).

The second condition in (4.1) above is trivially satisfied for k ∈ 1, 2. As we shall see, for these two cases and a FJA

Levy model, the stated necessary and sufficient condition (Bn/√hn → ∞) is the same as that needed to consistently

estimate the jump component process JT . This can be intuitively be explained in terms of how the two different

jump-misclassification errors affect the performance of the estimators (see Remark 4.8 below for further details).

For future comparisons, it is useful to report the associated asymptotics for k = 2:

E [TQV (X)[B]nT ]− Tσ2 = −2Tσφ

(Bn

σh1/2n

)(Bn

h1/2n

)+ Thn

(γ2 − λσ2

)+

2

3TλC0(f)B3

n + h.o.t. ,

Var (TQV (X)[B]nT ) = 2Tσ4hn +2

5TλB5

nC0(f) + h.o.t.

(4.4)

Remark 4.1. One important application of the asymptotic methods developed here is in designing objective criteria

for tuning the threshold parameter Bn. Indeed, a natural strategy is to choose Bn so that to minimize the leading order

terms of the MSE. Note that this may require stronger asymptotic representations than (4.4) since we are interested

in analyzing the leading order terms of the expressions therein relative to Bn. Thus, for instance, it is possible that in

the asymptotics for Var (TQV (X)[B]nT ), there is a term depending on Bn that is o(hn), but not necessarily o(B5n). So,

we need to make sure that the h.o.t.’s are of higher order relative to all the leading terms depending on Bn. A careful

inspection of the proof of Theorem 4.1 shows that this is the case for the asymptotics for E [TQV (X)[B]nT ]− Tσ2, but,

for Var (TQV (X)[B]nT ), we have:

Var (TQV (X)[B]nT ) = 2Tσ4hn +2

5TλB5

nC0(f)− 2Tσhnφ

(Bn

σ√hn

)(Bn

h1/2n

)3

+ h.o.t. (4.5)

Note that, indeed, though the third term on the right-hand side is o(hn), this is not necessarily o(B5n). Using (4.4) and

(4.5), we can write the following assymptotics for the MSE:

MSE = 2Tσ4hn +2

5TλB5

nC0(f)− 2Tσhnφ

(Bn

σ√hn

)(Bn

h1/2n

)3

+ 4T 2σ2φ2

(Bn

σh1/2n

)(Bn

h1/2n

)2

+ h.o.t.

It is then natural to take Bn to minimize the leading order terms of the right-hand side above. This leads to the

following equation that the minimizer Bn must satisfy:

0 = 2TλB4nC0(f)− 6Tσhnφ

(Bn

σ√hn

)B2n

h3/2n

+ 2Tσhnφ

(Bn

σ√hn

)(Bn

h1/2n

)3Bnσ2hn

+ 8T 2σ2φ2

(Bn

σh1/2n

)Bnhn− 8T 2σ2φ2

(Bn

σh1/2n

)(Bn

h1/2n

)2Bnσ2hn

.

(4.6)

Since Bn/√hn → ∞, the second and fourth terms of (4.6) are negligible with respect to the third and fifth terms,

respectively, and, thus, we can write (4.6) as follows:

2TλB4nC0(f) = −2Tσhnφ

(Bn

σ√hn

)(Bn

h1/2n

)3Bnσ2hn

(1 + o(1)) + 8T 2σ2φ2

(Bn

σh1/2n

)(Bn

h1/2n

)2Bnσ2hn

(1 + o(1)) . (4.7)

11

Now, suppose that for some constant c ∈ [0,∞),

limn→∞

hnφ(

Bnσ√hn

)(Bnh1/2n

)3Bnhn

B4n

= limn→∞

φ(

Bnσ√hn

)h

3/2n

= c. (4.8)

Then, dividing (4.7) by B4n, taking the limit when n→∞, and rearranging terms, we get

limn→∞

φ2(

Bnσ√hn

)(Bnh1/2n

)2Bnhn

B4n

= limn→∞

φ2(

Bnσ√hn

)Bnh2

n

=λC0(f)

4T+

c

4Tσ∈ (0,∞).

However, this contradicts (4.8) since

limn→∞

φ(

Bnσ√hn

)h

3/2n

= limn→∞

φ(

Bnσ√hn

)B

1/2n hn

√Bnhn

=∞,

because Bn/√hn →∞. By extending the previous argument to any subsequence, we can conclude that c in (4.8) must

be ∞, which is

limn→∞

hnφ(

Bnσ√hn

)(Bnh1/2n

)3Bnhn

B4n

= limn→∞

φ(

Bnσ√hn

)h

3/2n

=∞.

Therefore, the first term in (4.6) is negligible with respect to the third term. We can next divide Eq. (4.6) by the third

term therein and conclude that

limn→∞

8T 2σ2φ2(

Bnσh

1/2n

)(Bnh1/2n

)2Bnσ2hn

2Tσhnφ(

Bnσ√hn

)(Bnh1/2n

)3Bnσ2hn

= limn→∞

4Tσφ(

Bnσh

1/2n

)Bn√hn

= 1. (4.9)

This is the same asymptotic behavior obtained in Figueroa-Lopez and Mancini (2018) for their optimal threshold

(namely, that minimizing the MSE). In particular, it was shown therein that (4.9) implies that

Bn ∼ σ√

2hn ln(1/hn), (n→∞),

and we recover the same result as in Figueroa-Lopez and Mancini (2018). Note that the criterion here is not the

same as in that paper since, here, we are choosing Bn to minimize the leading order terms of the MSE, while in the

aforementioned paper, the threshold that minimizes the MSE was considered.

One of the implications of Theorem 4.1 is that the bias of the realized thresholded variation of kth-order can be

made to converge to 0 at the best possible rate of O(hn) by choosing Bn to converge to 0 slow enough compared to√hn (so that the first term in (4.2) converges to 0 faster than hn), but faster than h

k/2(k+1)n . By contrast, as shown

by our next theorem, the bias of MPVs can never beat the order O(hn).

Theorem 4.2. Under the class of FJA Levy processes and a finite-horizon sampling scheme with Tn ≡ T , for some

fixed T ∈ (0,∞), the MPV estimator MPV (X)n[r]T defined in (3.4), with exponent vector r = (r1, . . . , rk), will converge

in mean-squared error to Tσr+ if and only if the following two properties hold:

(i) rmax := max1≤j≤k

rj < 2, (ii) g(`) > −1, for all ` = 1, 2, . . . , k, (4.10)

12

where

g(`) :=

k∑j=`

1rj−`+1+rj≥2

(1− rj−`+1 + rj

2

)≤ 0. (4.11)

Furthermore, under the previous consistency conditions, the asymptotic rate of convergence for the bias is given by:

E[MPV (X)

n[r]T

]− Tσr+ ∼ TL(r)h1−rmax/2

n , (n→∞), (4.12)

where

L(r) := |Imax|λσ(r+−rmax)E[|ζ1|rmax ]m−1rmax

, with |Imax| := #i ∈ 1, 2, · · · , k : ri = rmax. (4.13)

Additionally, the variance of the estimator converges to 0 with a rate given by:

Var(MPV (X)

n[r]T

)∼ TL(r)h1+g(`∗)

n , (n→∞), (4.14)

where `∗ := argmin1≤`≤kg(`) and L(r) is a constant depending on the model’s parameters and the exponent vector r.

Remark 4.2. From the definition of g(`) given in (4.11), the set of vectors r ∈ Rk+ satisfying the two conditions stated

in (4.10) form a convex polytope. The second condition guarantees that the second moment bias of the sequence of

estimators will tend to 0 in the limit. As it turns out an expansion of the second moment of the MPV estimator yields

two types of terms, overlapping and non-overlapping block terms (see (A.10)-(A.12) in Appendix A and the subsequent

text therein for full details). Under the condition rmax < 2 the limiting behavior of the second moment is determined

by the non-overlapping block terms if and only if the overlapping block terms converge to 0, which corresponds precisely

to the conditions placed on g(`) for ` = 1, 2, . . . , k.

For a fixed target value of p := r+, the results given in (4.12) and (4.14) suggest that, in order to formulate an

efficient estimator, one should choose r such that rmax is as small as possible while simultaneously aiming to maximize

g(`∗), which by definition cannot exceed 0. This begs the notion of utilizing progressively longer averaging blocks, or

windows, and smaller exponent coefficients as the sample size increases. In other words, by allowing the order or

dimension of r to depend on the sample size, i.e. r ≡ r(n) ∈ Rkn+ , for some non-decreasing sequence kn ≤ kn+1 ∞

as n → ∞, sharper estimation results can be obtained. However, due to the presence of L(r) and L(r) in the first

order asymptotics of the bias and variance, the choice of an ‘optimal’ sequence of parameters is unclear. For example,

given any fixed α ∈ (0, 1] define a sequence (kn)n where kn := bαMnc the naive choice of r = (p/kn, · · · , p/kn) ∈ Rkn+

apparently yields a sequence of MPV estimators, which do not even converge in mean due to the asymptotic growth

associated with the |Imax| term defined in (4.13). Conversely, by selecting a single exponent vector coefficient to be

larger than the rest, but still a decreasing function of the sample size, this issue can be circumvented. However, the

problem of how to choose an ‘optimal’ or even ‘super-efficient’ averaging window length dependent on the sample size

remains. Nonetheless, this is an interesting problem which could be further explored in the future.

Example 4.1. From Eqs. (4.13) and (A.21) in the proof, we can further conclude that, for the Bipower Variation

(BPV) estimator (i.e., k = 2 and r1 = r2 = 1), the bias, variance, and mean-squared error relative to the parameter

13

σ2 take the following expressions:

E [BPV (X)nT ]− Tσ2 ∼ 2TσλE[|ζ1|]m−11 h1/2

n =: TL((1, 1))h1/2n

Var(BPV (X)nT ) ∼ Tσ4(1 + 2m2

1 − 3m41) + 2σ2λ(1 +m2

1)E[ζ21 ] +

(λE[ζ2

1 ])2

m−41 hn =: TL((1, 1))hn,

E[∣∣BPV (X)nT − Tσ2

∣∣2] ∼ (T 2L((1, 1))2 + TL((1, 1)))hn,

where recall that m1 := E|Z| =√

2/π for a standard Gaussian random variable Z. In the case of r1 = · · · = rk = 2/k,

for k > 2, the constant L(r) simplifies as

L(r) = σ4(1− 2k) + σ4βkkβk + 1− 2β

−(k−1)k

βk − 1,

where βk := m4/k/m22/k. As it turns out, L(r) ≈ 4.9σ4, as k →∞.

Remark 4.3. From the above rates of convergence it is clear that the bias of the BPV estimators converge to 0 rather

slowly compared to higher order multi-power variation estimators. In general, the bias of MPV estimators can be

asymptotically reduced by increasing the order k, however, as alluded to in the previous remark, the sample size should

be taken into consideration when selecting the order. Furthermore, as a consequence of (4.12), the bias can be made to

converge at a rate as close to hn as one wishes; however, the rate hn is not attainable by this class of estimators.

We now consider central limit theorems (CLT) for the class of TQV estimator introduced in (3.7). For comparison

purposes we present two important results from Mancini (2003) (see also Theorems 1 and 2 in Mancini (2009)).

Theorem 4.3 (Mancini (2003)). Let (Bn)n be a sequence of real numbers such that

limn→∞

Bn = 0 and limn→∞

Bn√hn log(1/hn)

=∞. (4.15)

Then, the following assertions hold true in a finite-horizon sampling scheme with Tn ≡ T :

1. For P-almost every ω ∈ Ω, there exists an N(ω) > 0 large enough such that

1∆ni N=0(ω) = 1|∆n

i X|≤Bn(ω),

for all i = 1, . . . ,Mn and all n ≥ N(ω). In particular, it follows that TQV (X)[B]nT →P∫ T

0σ2sds, as n→∞.

2. The asymptotic law of the TQV estimation error is given by:

h−1/2n

(TQV (X)[B]nT −

∫ T

0

σ2sds

)⇒D N

(0, 2

∫ T

0

σ4sds

)(n→∞). (4.16)

The following result relaxes the condition of Theorem 4.3 for the validity of the CLT.

14

Theorem 4.4. (CLT for TQV Estimators) Let B = (Bn)n be a threshold sequence satisfying the following condi-

tions:

(i) limn→∞

Bn√hn

=∞, (ii) limn→∞

Bn√hn

Mn∑i=1

σn,iφ

(Bnσn,i

)= 0. (4.17)

Then, under a finite-horizon sampling scheme with Tn ≡ T ,

h−1/2n

(TQV (X)[B]nT −

∫ T

0

σ2sds

)⇒D N

(0, 2

∫ T

0

σ4sds

), (n→∞). (4.18)

Remark 4.4. A sufficient condition for the validity of (4.17-ii) is given by

limn→∞

Bnhn

φ

(Bn

σT√hn

)= 0. (4.19)

It is an easy exercise to show that any threshold sequence satisfying (4.15) necessarily satisfies (4.19), but not the

other way. Therefore, Theorem 4.4 offers an extension to Theorem 2 of Mancini (2009) within the class of FJA

additive processes. An important instance of a sequence satisfying (4.19), but not (4.15), is Ban :=√

2σ2hn log(1/hn),

for σ ≥ σT , which was studied in Figueroa-Lopez and Nisen (2013) and Figueroa-Lopez and Mancini (2018) as a

byproduct of optimal thresholding selection methods.

Remark 4.5. The CLT stated in Theorem 4.4 can be easily be extended to the case where γ, σ, and λ are cadlag

stochastic processes independent of W , N , and ζjj≥1 and satisfying (2.2) and (4.17-ii) almost surely. Indeed, as

described in the proof of Theorem 4.4, the result is a consequence of the analogous result for the continuous component

Xc of X (which is valid in the more general framework considered here, in light of Theorem 1 in Barndorff-Nielsen

and Shephard (2007)) and the convergence to 0, in probability, of the following terms:

h−1/2n

Mn∑i=1

|∆ni X

c|21[∆ni N 6=0], h−1/2

n

Mn∑i=1

|∆ni X

c|21[|∆ni X

c|>Bn], h−1/2n

Mn∑i=1

|∆ni X|21[|∆n

i X|≤Bn,∆ni N 6=0].

To show the referred limits, it suffices to condition on G := σ(σt, γt, λt : t ≤ T ) and use dominated convergence. It is

also not hard to see that (4.17-ii) can be replaced by either of the following conditions:

limn→∞

Bn√hn

Mn∑i=1

E[σn,iφ

(Bnσn,i

)]= 0, lim

n→∞

Bnhn

E[σTφ

(Bn

σT√hn

)]= 0.

In order to develop a feasible version of the above central limit theorem, note that, by using a 4th order TPV

estimator3 with a threshold sequence satisfying the conditions limn→∞Bn/h1/2n = ∞ and limn→∞B9

n/h2n = 0, an

application of Slutsky’s Theorem implies

h−1/2n

(TQV (X)[B]nT −

∫ T0σ2sds)

√2TPV (X)[B]n,4T

⇒D N(0, 1), (n→∞).

3In fact, any consistent estimator of∫ T0 σ4

sds will suffice; e.g., a MPV estimator with r+ = 4.

15

4.2 Jump Component Estimators

In this section we study the performance of the jump component estimators of J and N . Intuitively, the choice of the

threshold sequence does not affect the recovery of the jumps’ sizes as strongly as it does the recovery of the corresponding

jumps’ times. This assertion will be formally justified below by showing that the former type of estimation requires

milder constraints on the set of threshold sequences. All the proofs in this section are presented in Appendix B.

Our first result examines the mean-squared consistency of the estimator N [B]nt defined in (3.12). As a byproduct,

we conclude necessary and sufficient conditions on the rate of convergence of the threshold sequence in order to achieve

mean-squared consistency. The following result is related to Theorem 3 in Mancini (2004), where it is proved that

P(N [B]nTn 6= NTn

)→ 0, as n→∞, for a threshold sequence of the form Bn = α

√hn ln(1/hn), but under the strong

assumption that the jumps are bounded away from the origin – recall it is assumed limn→∞Bn = limn→∞ hn = 0.

Proposition 4.5 (Consistent estimation of the Poisson process). Suppose that the jump density f satisfies

(2.3)-(2.4) for some bounded functions f+, f−, and that hn → 0 and Bn → 0. Then, the following assertions hold:

(1) The mean-squared error of the estimator N [B]nTn in (3.12) relative to the underlying Poisson process NTn con-

verges to 0 whenever

limn→∞

TnσTn√hnBn

φ

(Bn

σTn√hn

)= limn→∞

TnBnλTn = limn→∞

hnγTnBn

= limn→∞

BnγTnσ2Tn

= limn→∞

Tnhnλ2Tn = 0. (4.20)

In particular, in the finite time horizon estimation framework, all the conditions above reduce to

limn→∞

1√hnBn

φ

(Bn

σT√hn

)= 0. (4.21)

(2) In the FJA Levy model, the first two conditions in (4.20) imply the rest and these two are also necessary for the

MSE-consistency of N [B]nTn . In that case, the MSE of N [B]nTn admits the following decomposition:

E[∣∣∣N [B]nTn −NTn

∣∣∣2] = V (1)n + V (2)

n + h.o.t. (4.22)

where

V (1)n =

2σTn√hnBn

φ

(Bn

σ√hn

), V (2)

n = 2λTnBnC0(f). (4.23)

Remark 4.6. The terms V(1)n and V

(2)n in the mean-squared error decomposition (4.22) correspond to the leading terms

of the expected number of Type I and Type II errors, respectively. The necessity of the first (resp., second) condition

stated in (4.20) implicitly specifies how fast (resp., how slow) the threshold sequence can converge to zero and still allow

the corresponding estimators to consistently estimate the total number of jumps over the considered time horizon. In

particular, note that (Bn)n must satisfy that√hn Bn T−1

n , in the FJA Levy case.

Remark 4.7. The proof of Proposition 4.5 also yields the following non-asymptotic upper bound for the MSE:

E[∣∣∣N [B]nTn −NTn

∣∣∣2] ≤ V (1)n + V (2)

n +(V (1)n + V (2)

n + R(1)n

)2

+ R(2)n ,

16

where, for any ε ∈ (0, 1),

0 ≤ V (1)n ≤ 2

Tnhn

1[Bn(1−ε)≤γTnhn] +2

ε

TnσTn√hnBn

φ

(Bn

σTn√hn

)exp

(BnγTnσ2Tn

),

0 ≤ V (2)n ≤ ‖f‖∞λTnTnBn,

∣∣∣R(`)n

∣∣∣ ≤ 4λ2TnTnhn, ` = 1, 2.

In particular, if γ, σ, and λ were stochastic, but independent of W , N , and ζjj≥1, then sufficient conditions for the

MSE consistency of N [B]nTn are given by the following limits, as n→∞,

T 2nB

2nE[λ2Tn

]→ 0, TnhnE

[λ2Tn

]→ 0,

T 2n

hnBnE [γTn ]→ 0,

T 2n

hnB2n

E[σ2Tn exp

(− B2

n

σ2Tnhn

)exp

(2BnγTnσ2Tn

)]→ 0,

which correspond to the convergence of the second moments of V(`)n and R

(1)n and the first moment of R

(2)n .

Our next result is the analog of Proposition 4.5 but for the estimator J [B]nTn of the compound Poisson process JTn .

This result is related to Theorem 3 in Mancini (2009), where a CLT for J [B]nTn is established under the much stricter

condition (1.1), but in a more general Ito semimartingale framework.

Proposition 4.6 (Consistent estimation of the compound Poisson process). Suppose that the jump density f

satisfies (2.3)-(2.4) for some bounded functions f+, f−, and that hn → 0 and Bn → 0. Then, the following holds:

(1) The mean-squared error of the estimator J [B]nTn in (3.12), relative to the underlying compound Poisson process

JTn , converges to 0 provided that the conditions below are satisfied:

limn→∞

Tnhnλ2Tn = 0, lim

n→∞TnhnλTn γTn = 0, lim

n→∞Tnh

2nλTn γ

2Tn = 0, lim

n→∞TnhnλTn σ

2Tn = 0. (4.24)

limn→∞

TnσTnBn√hn

φ

(Bn

σTn√hn

)= 0, lim

n→∞

TnγTnBn

σTn√hn

φ

(Bn

σTn√hn

)= 0, lim

n→∞

TnγTn σTn√hn

Bnφ

(Bn

σTn√hn

)= 0,

(4.25)

limn→∞

BnγTnσ2Tn

= 0, limn→∞

BnhnγTn

=∞, limn→∞

B2n

σ2Tnhn

=∞, limn→∞

TnλTnB2n = 0. (4.26)

In particular, in the finite time horizon framework, all the conditions (4.24)-(4.26) reduce to

limn→∞

Bn√hn

=∞. (4.27)

(2) In the FJA Levy case, the following are necessary and sufficient conditions for the mean-squared convergence of

the estimator J [B]nTn toward JTn :

(i) limn→∞

Bn√hn

=∞, (ii) limn→∞

TnBn√hn

φ

(Bn

σ√hn

)= 0, (iii) lim

n→∞TnB

2n = 0. (4.28)

Remark 4.8. It is clear that (4.21) implies (4.27) and, thus, it transpires that, in the finite-time-horizon case, the

conditions required to “recover” the jump component Jt and the volatility σ are weaker than those required to recover the

17

Poisson process Nt driving Jt. A heuristic explanation of this phenomenon can be drawn from the following insightful

decompositions:

N [B]nTn −NTn =

Mn∑i=1

1[∆ni N 6=0] −NTn +

Mn∑i=1

1[|∆ni X|>Bn,∆n

i N=0] −Mn∑i=1

1[|∆ni X|≤Bn,∆n

i N 6=0], (4.29)

J [B]nTn − JTn =

Mn∑i=1

(∆ni X)1[∆n

i N 6=0] − JTn +

Mn∑i=1

(∆ni X)1[|∆n

i X|>Bn,∆ni N=0] −

Mn∑i=1

(∆ni X)1[|∆n

i X|≤Bn,∆ni N 6=0],

(4.30)

1

TnTQV [B]nTn − σ

2 =

(1

Tn

Mn∑i=1

|∆ni X|21[∆n

i N=0] − σ2

)+

1

Tn

Mn∑i=1

|∆ni X|21[|∆n

i X|≤Bn,∆ni N 6=0]

− 1

Tn

Mn∑i=1

|∆ni X|21[|∆n

i X|>Bn,∆ni N=0]. (4.31)

The relationships (4.29)-(4.31) suggest that the estimator N [B]n is “highly” sensitive to both types of jump misclas-

sifications. By contrast, the estimators J [B]n and TQV [B]nTn are far less sensitive to Type II errors. The crucial

difference being the presence of a ∆ni X term when a Type II error occurs together with the fact that limn→∞∆n

i Xc = 0

a.s. The above MSE decompositions also indicate that, in order to control Type II errors, it suffices that the threshold

sequence converge to zero, irrespective of its rate. By contrast, in order to control Type I errors, or spurious jump

detections, the threshold sequence cannot tend to zero too quickly. In other words, in demanding good jump detection

properties of our estimators, recovery of the jump component and the diffusion parameter comes for free, even though,

the converse is not true.

5 Conclusions

In this work, we obtain sufficient conditions for the mean-squared consistency of some threshold-based estimators for

the jump component JT , the total number of jumps NT , and the kth-order variation∫ T

0σksds of an additive process X

with finite jump activity. All the results can be extended to the class of conditionally additive processes and to provide

non-asymptotic upper bounds for the MSE of the estimators, similarly as it is done in Remark 4.7 for the estimator

of NT . Furthermore, if our results were to cover the simplest case of constant drift, volatility, and jump intensity,

the stated conditions will also be necessary and, thus, these are the best possible conditions. Precise asymptotic

decompositions of the bias and variance of the considered estimators are also provided. These in turn help to explain

how the different model parameters affect the performance of the estimators and the best possible rates of convergence.

Thus, for instance, by choosing Bn to converge to 0 slow enough compared to√hn (so that the first term in (4.2)

converges to 0 faster than hn), but faster than hk/2(k+1)n , the bias of the kth-order realized thresholded variation would

converge to 0 at the rate O(hn), which is the best possible. By contrast, as shown by our Theorem 4.2, the bias of

MPVs can never beat the order O(hn). Finally, we also give a new condition, weaker than those previously reported

in the literature, for the realized quadratic variation estimator to admit a feasible CLT.

18

Let us conclude with some remarks regarding possible extensions of our results to more general settings. There are

two main extensions to be considered: a general stochastic volatility model and infinite jump-activity. As mentioned

in the introduction, the results in the present paper can easily be extended to incorporate stochastic volatility as far

as the latter is independent of the Brownian and jump processes driving X. This can be done by conditioning on

the volatility process. However, if such an independence does not hold, the treatment would be much more technical.

The extension to infinite jump-activity will also require more technical considerations. However, there is some hope to

deal with these issues by using a similar approach to that of Figueroa-Lopez and Olafsson (2016). Namely, the idea

therein was to approximate the underlying process X with a process X satisfying simpler conditions (specifically, being

a Levy process) in such a way that functionals of X, say E[g(Xh)], are very close to the corresponding functional of X,

E[g(Xh)], as h→ 0. However, this direction is outside of the scope of the present work and is left for future research.

Acknowledgments: The first author’s research was partially supported by the NSF grants DMS-1561141 and DMS-

1613016. Both authors are thankful to two anonymous referees for many insightful and helpful comments.

A Proofs of Section 4.1

Throughout, Z denotes a standard normal random variable whose absolute moments are denoted by mr := E|Z|r.

Proof of Theorem 4.1. Let us start by noting the following useful decomposition, valid for even positive integers k,

E(|∆n

i X|k1[|∆ni X|≤Bn]

)= σkn,imk +R(1,k)

n,i +R(2,k)n,i −R

(3,k)n,i , (A.1)

where the remainder terms are given by

R(1,k)n,i = E

(|∆n

i X|k1[∆ni N=0]

)− σkn,imk =

(e−λn,i − 1

)σkn,imk + e−λn,i

k∑j=1

(k

j

)γjn,iσ

k−jn,i E

(Zk−j

),

R(2,k)n,i = E

(|∆n

i X|k1[|∆ni X|≤Bn,∆n

i N 6=0]

),

R(3,k)n,i = E

(|∆n

i X|k1[|∆ni X|>Bn,∆n

i N=0]

).

Using (2.2) and arguments alike those used in deriving (C.3) and (C.9), we have that

|R(1,k)n,i | ≤ C

(1)hk2 +1n , |R(2,k)

n,i | ≤ C(2)Bk+1

n hn, |R(3,k)n,i | ≤ C

(3)√hnB

k−1n φ

(Bn

σT√hn

), (A.2)

for universal constants C(`), independent of i and n, and depending only on k, σT , γT , λT , and ‖f‖∞. For completeness,

we check the bound for R(3,k)n,i . Let us start by nothing that

|R(3,k)n,i | ≤

k∑j=0

(k

j

)|γn,i|jσk−jn,i E

[|Z|k−j1[|γn,i+σn,iZ|>Bn]

]≤ 2

k∑j=0

(k

j

)γjT σ

k−jT h

k+j2

n E(Zk−j1[Z>un]

),

19

where we set un := Bn−γThnσT√hn

and assumed that n is large enough for un > 0 (recall that Bn √hn). By making a

change of variables, we have

E(Zm1[Z>un]

)=

2m/2−1

π1/2Γ

(m+ 1

2,u2n

2

)∼(

Bn

σT√hn

)m−1

φ

(Bn

σT√hn

), n→∞,

where Γ(s, x) :=∫∞xus−1e−udu is the upper incomplete gamma function and we used the well known asymptotic

property Γ(s, x) ∼ xs−1e−x, as x→∞. Finally, for n large enough, independently of i,

|R(3,k)n,i | ≤ 4

k∑j=0

(k

j

)γjT σ

k−jT h

k+j2

n

(Bn

σT√hn

)k−j−1

φ

(Bn

σT√hn

)= 4σTh

12nB

k−1n φ

(Bn

σT√hn

)(1 + o(1)) ,

where in the last equality we used that Bn √hn.

Now, we are ready to prove the result. Let us first analyze the bias of TPV (X)[B]n,kT , which for simplicity is denoted

by TPVn. Note that, from (A.1),

Bias (TPVn) =

n∑i=1

1

mkhk/2−1n

E[|∆n

i X|k1[|∆ni X|≤Bn]

]−∫ T

0

σkudu

=

n∑i=1

σkt∗i hn −∫ T

0

σkudu+ R(1)n + R(2)

n − R(3)n , (A.3)

where t∗i ∈ (ti−1, ti) is such that∫ titi−1

σ2udu = σ2

t∗ihn and, by (A.2), the reminder terms satisfy

|R(1)n | ≤ C(1)Thn, |R(2)

n | ≤ C(2)TBk+1n

hk2−1n

, |R(3)n | ≤ C(3)T

Bk−1n

hk−12

n

φ

(Bn

σT√hn

),

for some constants C`. It is now clear that the conditions given in (4.1) are sufficient for the bias to vanish. For ε > 0,

let δ > 0 be such that |σkv − σku| < ε whenever |u− v| < δ. Then, for n large enough,∣∣∣∣∣n∑i=1

σkt∗i hn −∫ T

0

σkudu

∣∣∣∣∣ ≤n∑i=1

∫ ti

ti−1

|σku − σkt∗i |du ≤ Tε,

which shows that the first term on the right-hand side of (A.3) converges to 0. Now, we consider the variance of the

TPV estimator. Clearly, by the independence of the increments of X and the decomposition (A.1),

Var (TPVn) ≤ 1

m2kh

k−2n

n∑i=1

E[|∆n

i X|2k1[|∆ni X|≤Bn]

]≤ R(0)

n + R(1)n,i + R(2)

n,i,

where the terms R(`)n are given by

R(0)n =

m2k

m2kh

k−2n

n∑i=1

(∫ ti

ti−1

σ2udu

)k, R(`)

n,i =1

m2kh

k−2n

n∑i=1

R(`,2k)n,i , ` = 1, 2,

and, thus, by (A.2) and the fact that(∫ ti

ti−1σ2udu

)k≤ σ2k

T hkn, are such that

|R(0)n | ≤ C(0)Thn, |R(1)

n | ≤ C(1)Th2n, |R(2)

n | ≤ C(2)TB2k+1n

hk−2n

,

20

for some constants C`. Therefore, we conclude that the variance converges to 0 whenever the conditions in (4.1) are

satisfied.

Let us now show that, for a FJA Levy model, the conditions given in (4.1) are necessary. To this end, note that

Biasn := Bias(TPV (X)[B]n,kT ;Tσk

)is such that

Biasn = T

(1

mkhk/2n

E[|∆n

1X|k1[|∆n1X|≤Bn,∆n

1N=0]

]− σk

)+

T

mkhk/2n

E[|∆n

1X|k1[|∆n1X|≤Bn,∆n

1N 6=0]

]=: A(1)

n +A(2)n . (A.4)

Lemma C.2 implies that, as far as Bn → 0,

A(2)n ∼

2TλC0(f)

(k + 1)mk

Bk+1n

hk/2−1n

. (A.5)

Now, suppose that L := lim infn→∞Bn/√hn ∈ [0,∞). Then, as shown in the proof of Lemma C.5, for a subsequence

njj≥1, limj→∞A(1)nj exists and is different from 0. Moreover,

limj→∞

Bk+1nj

hk/2−1nj

= limj→∞

(Bnj

h1/2nj

)kBnjhnj = 0,

which implies that limj→∞A(2)nj = 0 and, hence, limj→∞ Biasnj = limj→∞A

(1)nj 6= 0, which contradicts that the MSE

vanishes as n→∞. So, hereafter, we assume that condition (4.1-i) holds and show the necessity of (4.1-ii). First, note

that

Var(TPV (X)[B]n,kT

)=

T

m2kh

k−1n

E[|∆n

1X|2k1[|∆n1X|≤Bn]

]− T

m2kh

k−1n

E[|∆n

1X|k1[|∆n1X|≤Bn]

]2=: D(1)

n −(D(2)n

)2

.

Next, by decomposing 1[|∆n1X|≤Bn] as 1[|∆n

1X|≤Bn,∆n1N 6=0] + 1[|∆n

1X|≤Bn,∆n1N=0] and applying Lemmas C.2 and C.5, it

follows that

D(1)n =

T

m2kh

k−1n

(hnλ

2B2k+1n

2k + 1C0(f)

)+

T

m2kh

k−1n

(hknσ2k2k

π1/2Γ

(2k + 1

2

))+ h.o.t.,

D(2)n =

T 1/2

mkhk−12

n

(hnλ

2Bk+1n

k + 1C0(f)

)+

T 1/2

mkhk−12

n

(hk/2n

σk2k/2

π1/2Γ

(k + 1

2

))+ h.o.t.

Since the second terms in the decompositions of D(1)n and D(2)

n converge to 0, regardless of Bn, and

B2k+2n

hk−3n

=B2k+1n

hk−2n

Bnhn B2k+1n

hk−2n

,

we conclude that, for the variance to vanish, it is necessary that condition (4.1-ii) holds true.

Finally, from the previous decomposition for the variance, the asymptotic decomposition (4.3) holds. While (4.2)

is deduced from the decomposition (A.4) and the asymptotic property (A.5) and the following easy consequence of

Lemma C.5:

A(1)n = − Tπ1/2

2k/2−1Γ(k+1

2

) Bk−1n

h(k−1)/2n

φ

(Bnσn

)+ TΛ(1)

n hn + h.o.t.

21

Proof of Theorem 4.2. We prove the result through three steps:

Step 1. Here, we show that the estimator MPV (X)n[r]T is an asymptotically unbiased estimator for Tσr+ if and only

if rmax < 2 and that the asymptotic behavior of the bias is given as in (4.12). Note that, by conditioning on ∆ni N ,

E[|∆ni X|ri ] = h

ri/2n e−λnσrimri +Rn(ri), where

Rn(s) := e−λnE [|γn + σ∆niW |s]− e−λnE [|σ∆n

iW |s] + E[|γn + σ∆n

iW + ∆ni J |

s1[∆n

i N 6=0]

], (A.6)

and from Lemmas C.7 and C.8, Rn(r) ∼ λhnE[|ζ1|r], as n → ∞. Let a(n)i := h

ri/2n e−λnσrimri and b

(n)i := Rn(ri).

Using the fact that the increments are i.i.d., we may then express the first moment as follows:

E[MPV (X)n[r]T ] =

h1−r+/2n

C(r)(Mn − k + 1)

k∏i=1

(a

(n)i + b

(n)i

)=h

1−r+/2n

C(r)(Mn − k + 1)

2k∑j=1

k∏i=1

c(n)i,j , (A.7)

where c(n)i,j ∈ a

(n)i , b

(n)i for each fixed j. It is then clear that there exists a term in the expansion (A.7) of the order

h1−rmax/2n , which would not converge to 0, as n → ∞, if rmax ≥ 2. In turn, this shows the necessity of the condition

rmax < 2 for the bias to converge to 0, since all of the terms in the expansion (A.7) are positive for n large enough. To

show (4.12), let us observe that, under the condition rmax < 2, the leading order term corresponds to∏ki=1 a

(n)i , which

can be written as

Tσr+ + T (e−kλn − 1)σr+ − T (k − 1)

Mnσr+e−kλn = Tσr+ +O(hn).

To find the rate of convergence of the bias, consider Imax := i ∈ 1, 2, · · · , k : ri = rmax and let |Imax| denotes its

cardinality. Then, the bias Hn is such that

Hn ∼ T∑

i′∈Imax

h− r+

2n

C(r)

Mn − k + 1

Mnb(n)

i′×∏i 6=i′

a(n)i ∼ T |Imax|λσ(r+−rmax)E[|ζ1|rmax ]m−1

rmaxh

1− rmax2

n , (A.8)

as n→∞, which justifies (4.12).

Step 2. Throughout this step, we assume that rmax < 2, and show that the condition (4.10-ii) is necessary and

sufficient for

Bias

(∣∣∣MPV (X)n[r]T

∣∣∣2 ;T 2σ2r+)→ 0. (A.9)

We begin by considering the following obvious decomposition:

∣∣∣MPV (X)n[r]T

∣∣∣2 = S(1)n + 2

k∑`=2

S(`)n + 2S(k+1)

n , (A.10)

where, with the convention that 1 =∏kj=k+1,

S(`)n :=

h2−r+n

C(r)2

Mn−k−`+2∑i=1

`−1∏j=1

|∆i+j−1X|rjk∏j=`

|∆i+j−1X|rj−`+1+rj

k+`−1∏j=k+1

|∆i+j−1X|rj−`+1 ; ` = 1, . . . , k, (A.11)

S(k+1)n :=

h2−r+n

C(r)2

Mn−2k+1∑i=1

Mn−k+1∑m=k+i

k∏j=1

|∆i+j−1X|rjk∏q=1

|∆m+q−1X|rq . (A.12)

22

From here it is important to note that the right-hand side of (A.10) consists of two types of components or terms.

Namely, those corresponding to overlapping blocks, as defined in (A.11), and a non-overlapping block term given in

(A.12). As it turns out, under the condition rmax < 2 the non-overlapping block component will converge in mean

to the target value T 2σ2r+/2. Hence, in order to ensure the second moment bias of the sequence of MPV estimators

converges to 0 in the limit the expected value of the overlapping block components necessarily needs to vanish.

Note that each overlapping block component, S(`)n , is the sum of Mn − k + 2 − ` identically distributed terms for

` = 1, . . . , k, while the non-overlapping block component, S(k+1)n , contains

(Mn−2k+2

2

)such terms. Let Rn(s) be defined

as in (A.6) and recall that Rn(s) ∼ λhnE[|ζ1|s], as n → ∞. An analysis of the expected value of the non-overlapping

block term shows that

E[S(k+1)n

]=h2−r+n

C(r)2

(Mn − 2k + 2

2

) k∏j=1

[e−λnhrj/2n σrjmrj +Rn(rj)

]2(A.13)

=T 2h−r

+

n

C(r)2

(Mn − 2k + 2)(Mn − 2k + 1)

2M2n

k∏j=1

[e−2λnhrjn σ

2rjm2rj + 2e−λnhrj/2n σrjmrjRn(rj) +Rn(rj)

2]

=T 2σ2r+

2+

3− 4k

Mn

T 2σ2r+

2+Rn + o(h1−rmax/2

n ), where Rn = C(k+1)r h1−rmax/2

n , (n→∞), (A.14)

and C(k+1)r := 2T 2|Imax|λσ(2r+−rmax)E [|ζ|rmax ]m−1

rmaxand |Imax| := #i ∈ 1, 2, · · · , k : ri = rmax. We are now ready

to show that (4.10-ii) is necessary and sufficient for (A.9) to hold. First, from (A.14), it transpires that if rmax < 2,

then limn→∞ E[S

(k+1)n

]= T 2σ2r+/2. Therefore, for (A.9) to be satisfied, it suffices to show that

limn→∞

E[S(`)n

]= 0, ` = 1, 2, · · · , k. (A.15)

The necessity of the above condition is also clear since each overlapping block term S`n is nonnegative. To show that

the condition (4.10-ii) is equivalent to (A.15), let us note that

E[S(`)n

]= T 2 h

−r+n

C(r)2

Mn − k − `+ 2

M2n

`−1∏j=1

(e−λnhrj/2n σrjmrj +Rn(rj)

)×

k∏j=`

(e−λnh

(rj−`+1+rj)/2n σ(rj−`+1+rj)m(rj−`+1+rj) +Rn(rj−`+1 + rj)

)× (A.16)

k+`−1∏j=k+1

(e−λnh

rj−`+1/2n σrj−`+1mrj−`+1

+Rn(rj−`+1)),

whose slowest-converging term takes the form

T 2 h−r+n

C(r)2

1

Mn

`−1∏j=1

hrj2n σrjmrj

k∏j=`

h1∧

rj−`+1+rj2

n kj,`(r)

k+`−1∏j=k+1

hrj−`+1

2n σrj−`+1mrj−`+1

,

23

where kj,`(r) = σ(rj−`+1+rj)m(rj−`+1+rj)1(rj−`+1+rj)/2≤1 + λE|ζ1|rj−`+1+rj1(rj−`+1+rj)/2≥1. It is now clear that this

term will vanish if and only if g(`) > −1, where

g(`) :=

`−1∑j=1

rj2

+

k∑j=`

[rj−`+1 + rj

2∧ 1

]+

k+`−1∑j=k+1

rj−`+1

2− r+; ` = 1, 2, . . . , k.

The above function can be simplified to the expression given in (4.11). For future reference, let us also remark that,

under the conditions given in (4.10),

E[S(`)n

]∼ TC(`)

r h1+g(`)n (n→∞), ` = 1, . . . , k, (A.17)

where

C(`)r =

1

C(r)2

`−1∏j=1

σrjmrj

k∏j=`

kj,`(r)

k+`−1∏j=k+1

σrj−`+1mrj−`+1. (A.18)

Step 3. Let us first note that Step 1 and 2 above imply that first assertion of the theorem. Indeed, if both conditions

in (4.10) are satisfied, then it was proved that

Bias(∣∣∣MPV (X)

n[r]T

∣∣∣ ;Tσr+)→ 0, Bias

(∣∣∣MPV (X)n[r]T

∣∣∣2 ;T 2σ2r+)→ 0, n→∞, (A.19)

which, as it is well known, implies that MSE(∣∣∣MPV (X)

n[r]T

∣∣∣ ;Tσr+)→ 0. Reciprocally, if the latter limit holds, then

the first limit in (A.19) must hold since MSE ≥ Bias2. As shown in Step 1, if the bias of the MPV estimator vanishes,

then rmax < 2, and, as shown in Step 2 above, we will then have that the condition (4.10-ii) holds as well.

The asymptotics (4.12) was shown in the Step 1 above. Therefore, the only remaining assertion to prove is the

asymptotics (4.14). To that end, let us first recall from (A.7), (A.10), and (A.13) that the first and second moment of

the MPV estimator are given by

E[MPV (X)

n[r]T

]=h

1−r+/2n

C(r)(Mn − k + 1)

k∏j=1

(e−λnhrj/2n σrjmrj +Rn(rj)

)

E[∣∣∣MPV (X)

n[r]T

∣∣∣2] =2h2−r+

n

C(r)2

(Mn − 2k + 2

2

) k∏j=1

[e−λnhrj/2n σrjmrj +Rn(rj)

]2+ E

[S(1)n

]+ 2

k∑`=2

E[S(`)n

],

where S(`)n `=1,2,...,k are given as in (A.10). From these representations, it follows that

V ar(MPV (X)

n[r]T

)= T 2Dn,k

h−r+

n

C(r)2

k∏j=1

(e−λnσrjhrj/2n mrj +Rn(rj)

)2

+ E[S(1)n

]+ 2

k∑`=2

E[S(`)n

], (A.20)

where

Dn,k :=(Mn − 2k + 2)(Mn − 2k + 1)

M2n

− (Mn − k + 1)2

M2n

∼ (1− 2k)

Mn, as n→∞.

The asymptotics of E[S

(`)n

]for ` = 1, . . . , k are stated in (A.17), while the asymptotics of the first term in (A.20) is

given by (see also (A.13))

T 2Dn,kh−r

+

n

C(r)2

k∏j=1

(e−λnσrjhrj/2n mrj +Rn(rj)

)2

= T (1− 2k)σ2r+hn + o(hn).

24

Therefore, we deduce the asymptotic behavior (4.14) for Var(MPV (X)

n[r]T

)with a constant L(r) given by

L(r) := 10<rmax≤1σ2r+(1− 2k) + C(1)

r + 2

k∑`=2

C(`)r , (A.21)

where the constants C(`)r , ` = 1, 2, . . . , k, are defined by (A.18).

Proof of Theorem 4.4. Let us begin by noting that

h−1/2n

Mn∑i=1

(|∆n

i X|21[∆ni N=0] − σ2

n,i

) n→∞=⇒ D N

(0, 2

∫ T

0

σ4sds

). (A.22)

The limit (A.22) can be derived from Theorem 1 in Barndorff-Nielsen and Shephard (2007), which states that

h−1/2n

Mn∑i=1

(|∆n

i Xc|2 − σ2

n,i

) n→∞=⇒ D N

(0, 2

∫ T

0

σ4sds

), (A.23)

and the fact that Rn := h−1/2n

∑Mn

i=1 |∆ni X

c|21[∆ni N 6=0]

n→∞→P 0 since

E [Rn] = h−1/2n

Mn∑i=1

(1− e−λn,i

) (γ2n,i + σ2

n,i

)≤ h1/2

n T λT(γ2Thn + σ2

T

) n→∞−→ 0.

Next, we show that, under the conditions given in (4.17),

h−1/2n

(Mn∑i=1

|∆ni X|21[|∆n

i X|≤Bn,∆ni N=0] −

∫ T

0

σ2sds

)n→∞=⇒ D N

(0, 2

∫ T

0

σ4sds

). (A.24)

To this end, we show that

Tn :=

Mn∑i=1

|∆ni X|21[|∆n

i X|>Bn,∆ni N=0] = oP (h1/2

n ).

Since Tn is nonnegative, it suffices to show that h−1/2n ETn → 0, as n→∞. Using (2.7),

E [Tn] =

Mn∑i=1

e−λn,i(σ2n,i + γ2

n,i

)(Φ

(Bn + γn,iσn,i

)+ Φ

(Bn − γn,iσn,i

))(A.25)

+

Mn∑i=1

e−λn,iσn,i

((Bn − γn,i)φ

(Bn + γn,iσn,i

)+ (Bn + γn,i)φ

(Bn − γn,iσn,i

)).

Applying (2.6) and after some simplifications, for large enough n,

E [Tn] ≤Mn∑i=1

σn,i(B2n + σ2

n,i

)( 1

Bn + γn,iφ

(Bn + γn,iσn,i

)+

1

Bn − γn,iφ

(Bn − γn,iσn,i

)). (A.26)

Finally, using that |γn,i| ≤ γThn and σ2Thn ≤ σ2

n,i ≤ σ2Thn, one can see that the right-hand side of (A.26) multiplied

by h−1/2n converges to 0 if and only if (4.17) holds.

We are now ready to show (4.16). We begin with the decomposition

TQV (X)[B]nT =

Mn∑i=1

|∆ni X|21[|∆n

i X|≤Bn,∆ni N=0] +

Mn∑i=1

|∆ni X|21[|∆n

i X|≤Bn,∆ni N 6=0] =: D(1)

n +D(2)n .

25

From (A.24) and Slutsky’s Theorem, it is clear that it suffices to show that h−1/2n D

(2)n

n→∞→P 0. For the latter identity,

let us consider the following:

P

(1

h1/2n

Mn∑i=1

|∆ni X|21[|∆n

i X|≤Bn,∆ni N 6=0] 6= 0

)≤ P

(Mn⋃i=1

[|∆ni X| ≤ Bn,∆n

i N 6= 0]

)

= 1− exp

(Mn∑i=1

ln (1− P (|∆ni X| ≤ Bn,∆n

i N 6= 0))

)

≤ 1− exp

(Mn∑i=1

ln(1− 2‖f‖∞BnhnλT

))(A.27)

where the last inequality follows from the next easy implications of (C.9):

P (|∆ni X| ≤ Bn,∆n

i N 6= 0) = e−λn,i∞∑k=1

P

∣∣∣∣∣∣σn,iZ + γn,i +

k∑j=1

ζj

∣∣∣∣∣∣ ≤ Bn λkn,i

k!

≤ 2Bne−λn,i

∞∑k=1

‖f∗k‖∞λkn,ik!

≤ 2‖f‖∞Bnλn,i.

It is now clear that, provided that Bn → 0, (A.27) tends to 0, which in turn implies that limn→∞

P(h−1/2n D(2)

n 6= 0)

= 0.

The proof is now complete.

B Proofs of Section 4.2

Proof of Proposition 4.5. Throughout, Z denotes a standard normal variable. We fix Ai := 1[|∆ni X|>Bn] −∆n

i N ,

for i = 1, 2, . . . ,Mn, and note that (Ai)Mni=1 form a collection of independent random variables. We begin by recording

the first and second moments of each Ai. By conditioning on ∆ni N ,

E[Ai] = e−λn,iP (|γn,i + σn,iZ| > Bn)− λn,iP (|γn,i + σn,iZ + ζ1| ≤ Bn) +R(1)n,i,

E[A2i ] = e−λn,iP (|γn,i + σn,iZ| > Bn) + λn,iP (|γn,i + σn,iZ + ζ1| ≤ Bn) +R

(2)n,i,

where

R(1)n,i := λn,iP (|γn,i + σn,iZ + ζ1| > Bn) (e−λn,i − 1) +

∞∑j=2

e−λn,iλjn,ij!

P (|λn,i + σn,iZ + ζ1 + · · ·+ ζj | > Bn)

R(2)n,i := λn,iP (|γn,i + σn,iZ + ζ1| > Bn) (1− e−λn,i) + λ2

n,i +

∞∑j=2

e−λn,iλjn,ij!

(1− 2j)P (|γn,i + σn,iZ + ζ1 + · · ·+ ζj | > Bn) .

26

Note that∣∣∣R(1)

n,i

∣∣∣ ≤ 2λ2n,i and

∣∣∣R(2)n,i

∣∣∣ ≤ 4λ2n,i. Next, using the mutual independence of the Ai’s,

E[∣∣∣N [B]nTn −NTn

∣∣∣2] =

Mn∑i=1

E[A2i ] +

∑i6=j

E[Ai]E[Aj ]

≤Mn∑i=1

E[A2i ] +

(Mn∑i=1

E[Ai]

)2

= V (1)n + V (2)

n +(V (1)n + V (2)

n + R(1)n

)2

+ R(2)n ,

where we have fixed

V (1)n :=

Mn∑i=1

e−λn,iP (|γn,i + σn,iZ| > Bn) , V (2)n :=

Mn∑i=1

λn,iP (|γn,i + σn,iZ + ζ1| ≤ Bn) ,

R(1)n :=

Mn∑i=1

R(1)n,i, R(2)

n :=

Mn∑i=1

R(2)n,i.

Note that, for ` = 1, 2 and some constants K`,∣∣∣R(`)n

∣∣∣ ≤ K`

Mn∑i=1

λ2n,i ≤ K`hnTnλ

2Tn → 0, n→ 0,

which converges to 0 due to the last limit in (4.20). Thus, for the MSE of N [B]nt to converges to 0, it suffices that

limn→∞ V(1)n = limn→∞ V

(2)n = 0. From (C.9),

V (2)n ≤ 2‖f‖∞Bn

Mn∑i=1

λn,i ≤ 2‖f‖∞BnλTnTn,

which converges to 0 by the second limit in (4.20). Now, from the third condition in (4.20) and the Gaussian tail

estimate (2.6),

V (1)n =

Mn∑i=1

e−λn,i(

Φ

(Bn + γn,iσn,i

)+ Φ

(Bn − γn,iσn,i

))≤ 2MnΦ

(Bn − γTnhnσTn√hn

)≤ 2

MnσTn√hn

Bn − γTnhnφ

(Bn

σTn√hn

)exp

(BnγTnσ2Tn

),

which converges to 0 by conditions in (4.20).

Now, we consider the second assertion dealing with a FJA Levy model. In that case, it is easy to see that

R(2)n = MnR

(2)n ∼Mnh

2nλ

2/2 > 0 whenever hn, Bn → 0. Therefore, for large enough n,

E[∣∣∣N [B]nTn −NTn

∣∣∣2] ≥ V (1)n + V (2)

n ≥ 0.

Thus, if the mean-squared error vanishes, then limn→∞ V(1)n = limn→∞ V

(2)n = 0. Now, by (C.10) with k = 0,

P(|γhn + σ

√hnZ + ζ1| ≤ Bn

)∼ 2BnC0(f) when n → ∞ and, hence, V

(2)n ∼ 2λMnhnBnC0(f), as n → ∞. In

27

particular, if V(2)n converges to 0, the second condition in (4.20) must hold. On the other hand, if V

(1)n converges to 0,

then necessarily limn→∞Bn/h1/2n =∞ and, by (2.6),

V (1)n ∼ 2Mnσ

√hn

Bnφ

(Bn

σ√hn

), (n→∞), (B.1)

which further implies that the first condition in (4.20) must hold. Finally, the decomposition (4.22) directly follows

from the asymptotic rates for V(1)n , V

(2)n , R

(1)n , and R

(2)n stated above and the next decomposition:

E[∣∣∣N [B]nTn −NTn

∣∣∣2] = V (1)n + V (2)

n +Mn(Mn − 1)

M2n

(V (1)n + V (2)

n + R(1)n

)2

+ R(2)n .

Proof of Proposition 4.6. Throughout we will assume γ 6= 0. The proof is similar and simpler if γ = 0. Let

Z1, Z2, . . . denote independent standard Gaussian variables and let Yi := ∆ni X1[|∆n

i X|>Bn]−∆ni J so that J [B]nTn−JTn =∑Mn

i=1 Yi. Note that ∆ni J = ∆n

i J1[∆ni N 6=0] and, thus,

Yi = (∆ni X)1[|∆n

i X|>Bn,∆ni N=0] − (∆n

i X)1[∆ni N 6=0,|∆n

i X|≤Bn] + (∆ni X

c)1[∆ni N 6=0]

=: T(1)n,i + T

(2)n,i + T

(3)n,i .

Hence,

E[∣∣∣J [B]nTn − JTn

∣∣∣2] =

Mn∑i=1

E[Y 2i ] +

∑i 6=j

E[Yi]E[Yj ] ≤Mn∑i=1

E[Y 2i ] +

(Mn∑i=1

E[Yi]

)2

(B.2)

≤ 3

3∑`=1

Mn∑i=1

E[(T

(`)n,i

)2]

+

(3∑`=1

Mn∑i=1

E[T

(`)n,i

])2

. (B.3)

Therefore, for the above expression to converge to 0, it suffices to show that

limn→∞

Mn∑i=1

E[(T

(`)n,i

)2]

= 0, limn→∞

Mn∑i=1

E[T

(`)n,i

]= 0, ` = 1, 2, 3.

First, using (2.7), we can find the first two moments of T(1)n,i as

E[T

(1)n,i

]= e−λn,iσn,i

(φ

(Bn − γn,iσn,i

)− φ

(Bn + γn,iσn,i

))+ e−λn,iγn,i

(Φ

(Bn + γn,iσn,i

)+ Φ

(Bn − γn,iσn,i

))E[(T

(1)n,i

)2]

= e−λn,i(σ2n,i + γ2

n,i

)(Φ

(Bn + γn,iσn,i

)+ Φ

(Bn − γn,iσn,i

))(B.4)

+ e−λn,iσn,i

((Bn − γn,i)φ

(Bn + γn,iσn,i

)+ (Bn + γn,i)φ

(Bn − γn,iσn,i

)).

Next, using (2.6), which can be applied in light of the second condition in (4.26), and some simplifications,

∣∣∣E [T (1)n,i

]∣∣∣ ≤ φ( Bnσn,i

)e

Bn|γn,i|

σ2n,i

Bn|γn,i|σn,i

+ |γn,i|σn,i(

1

Bn + γn,iφ

(Bn + γn,iσn,i

)+

1

Bn − γn,iφ

(Bn − γn,iσn,i

)).

28

Thus, using (2.2),

Mn∑i=1

∣∣∣E [T (1)n,i

]∣∣∣ ≤Mn

√hnφ

(Bn

σTn√hn

)exp

(BnγTnσ2Tn

)BnγTnσTn

+ 2MnγTn σTnh

3/2n

Bn − γTnhnφ

(Bn

σTn√hn

)exp

(BnγTnσ2Tn

),

which vanishes as n→∞ in light of the conditions in (4.25)-(4.26). Similarly, applying (2.6),

Mn∑i=1

E[(T

(1)n,i

)2]≤

Mn∑i=1

σn,i(B2n + σ2

n,i

)( 1

Bn + γn,iφ

(Bn + γn,iσn,i

)+

1

Bn − γn,iφ

(Bn − γn,iσn,i

))(B.5)

≤ 2MnσTn√hnB2n + σ2

Tnhn

Bn − γTnhnφ

(Bn

σTn√hn

)exp

(BnγTnσ2Tn

),

which again converges to 0 due to the conditions in (4.25)-(4.26). To handle T (2), let us fix ζk :=∑kj=1 ζj and note

that, by (C.9), for m = 1, 2,∣∣∣E[(T(2)n,i )m]

∣∣∣ ≤ e−λn,i ∞∑k=1

E(∣∣σn,iZ + γn,i + ζk

∣∣m 1[|σn,iZ+γn,i+ζk|≤Bn]

) λkn,ik!

(B.6)

≤ 2Bm+1n

m+ 1e−λn,i

∞∑k=1

‖f∗k‖∞λkn,ik!≤ 2‖f‖∞

Bm+1n

m+ 1λn,i.

Therefore, using (2.2),Mn∑i=1

∣∣∣E[(T(2)n,i )m]

∣∣∣ ≤ 2‖f‖∞Bm+1n

m+ 1

Mn∑i=1

λn,i ≤ 2‖f‖∞Bm+1n

m+ 1TnλTn ,

which vanishes due to the last condition in (4.26). Finally, we can directly compute

E[T(3)n,i ] = γn,i

(1− e−λn,i

), E[|T (3)

n |2] =(1− e−λn,i

) (γ2n,i + σ2

n,i

). (B.7)

Thus,Mn∑i=1

∣∣∣E[T(3)n,i ]∣∣∣ ≤ TnγTn λTnhn, Mn∑

i=1

E[(T (3)n )2] ≤ TnλTnhn

(γ2Tnhn + σ2

Tn

),

which vanishes due to the last three conditions in (4.24).

We now show the second assertion of the proposition and, thus, hereafter we assume a FJA Levy model and fix

γn = γhn, σn = σ√hn, and λn = λhn. In that case, it is easy to check that the three conditions in (4.28) imply all the

conditions in (4.24)-(4.26). To show the necessity of (4.28), it is useful to note that, in the FJA case, (B.2) can further

be simplified as follows:

E[|J [B]nTn − JTn |

2]

=

Mn∑i=1

E[Y 2i ] +

∑i 6=j

E[Yi]E[Yj ] = MnE[Y 21 ] +Mn(Mn − 1)E[Y1]2. (B.8)

By writing ζk =∑ki=1 ζi and conditioning on Nhn ,

E[Y 21 ] =

∞∑k=0

e−λnλknk!

E[(

γn + σnZ + ζk)1[|γn+σnZ+ζk|>Bn] − ζk

2]

(B.9)

Note that the first term in (B.9) (corresponding to k = 0) is precisely E[|T (1)n,1|2]. Therefore,

E[|J [B]nTn − JTn |

2]≥MnE[|T (1)

n |2] = MnE[|γn + σnZ|2 1[|γn+σnZ|>Bn,∆n

i N=0]

]≥ 0.

29

Observe that

E[|T (1)n |2] = e−λnσ2hn

[∫ ∞(Bn−γn)/σn

|γh1/2n /σ + z|2φ(z)dz +

∫ −(Bn+γn)/σn

−∞|γh1/2

n /σ + z|2φ(z)dz

].

Suppose that L := lim infn→∞Bn/h1/2n <∞ and let nk be an increasing subsequence such that limk→∞Bnk/

√hnk = L.

Then, from the Dominated Convergence Theorem,

limk→∞

E[|T (1)nk |2]

hnk= σ2

∫(−L/σ,L/σ)c

z2φ(z)dz,

which implies that limk→∞MnkE[|T (1)nk |2] > 0, since, by the assumption of the sampling design, limn→∞Mnhn ∈ (0,∞].

Therefore, limn→∞MSE(J [B]nTn ; JTn

)= 0 implies that limn→∞Bn/

√hn =∞. Furthermore, from Lemma C.1,

E[|T (1)n |2] ∼ 2σBnh

1/2n φ

(Bnσn

),

and, thus, the second limit in (4.28) must hold. To show that the last limit therein holds, consider the second term on

the right-hand side of (B.8) and note that we must have that MnE[Y1] → 0. Next, applying Lemmas C.1 and C.2 to

the decomposition

E[Y1] = E[(∆n1X1[|∆n

1X|>Bn] −∆n1J)1[∆n

1N=0]] + E[(∆n1X1[|∆n

1X|>Bn] −∆n1J)1[∆n

1N 6=0]]

= E[T(1)n,1] + E[∆n

1Xc1[∆n

1N 6=0]]− E[(∆n1X)1[|∆n

1X|≤Bn,∆n1N 6=0]],

we get:

E[Y1] = 2γ

σBnh

1/2n φ

(Bnσn

)+ γhn(1− e−λhn)− λhnB

2n

2C(f) + h.o.t.

Since Bn √hn and TnBnh

−1/2n φ

(Bnσn

)→ 0, we must have that the last limit in (4.28) holds.

C Supporting Technical Lemmas

Throughout this section, X denotes a FJA Levy process with constant drift γ, volatility σ, and jump intensity λ. Let

us recall the notation γn := γhn, σ2n := σ2hn, and λn = λhn. In order to unify the presentation, we shall use the

following terminology for a given threshold sequence B = (Bn)n:

LIn[B](k) := E[(∆n

1X)k1[|∆n1X|>Bn,∆n

1N=0]

], LIIn [B](k) := E

[(∆n

1X)k1[|∆n1X|≤Bn,∆n

1N 6=0]

], (C.1)

LII,Absn [B](k) := E[|∆n

1X|k1[|∆n1X|≤Bn,∆n

1N 6=0]

], GIn[B](k) := E

[|∆n

1X|k1[|∆n1X|≤Bn,∆n

1N=0]

]. (C.2)

Lemma C.1. Let B = (Bn)n be a threshold sequence such that limn→∞Bnh−1/2n =∞. Then, as n→∞,

LIn[B](k) ∼

2σnB

k−1n φ

(Bnσn

); k ∈ 0, 2, 4, . . . ,

2σnBknγσ2φ

(Bnσn

); k ∈ 1, 3, 5, . . . ,

(C.3)

Furthermore, if γ = 0 and k is an odd positive integer, then LIn[B](k) = 0.

30

Proof. Throughout, Z denotes a standard normal random variable. We begin by assuming γ 6= 0 and by noticing that

LIn[B](k) = e−λnk∑j=0

(k

j

)γjnσ

k−jn E

[Zk−j1[|γn+σnZ|>Bn]

]. (C.4)

Let G(m)n (B) := E

[Zm1[|γn+σnZ|>Bn]

]and note that

G(m)n (B) =

∫ ∞Bn−γnσn

zmφ(z)dz +

∫ − (Bn+γn)σn

−∞zmφ(z)dz.

Let un := (Bn − γn)/√

2σ2n and un := (Bn + γn)/

√2σ2

n and observe that, since limn→∞ un = limn→∞ un = +∞, for

n sufficiently large, minun, un > 0. In that case, by making a change of variables, we have

G(m)n (B) =

2m/2−1

π1/2Γ

(m+ 1

2, u2n

)+ (−1)m

2m/2−1

π1/2Γ

(m+ 1

2, u2n

),

where Γ(s, x) :=∫∞xus−1e−udu is the upper incomplete gamma function. In the case that m ∈ 0, 2, 4, . . . , the well

known asymptotic property Γ(s, x) ∼ xs−1e−x, as x→∞, implies that, as n→∞,

G(m)n (B) ∼ 2

(Bnσn

)m−1

φ

(Bnσn

). (C.5)

For m ∈ 1, 3, 5, . . . , note that

G(m)n (B) =

2m/2−1

π1/2

[∫ ∞u2n

w(m−1)/2e−wdw −∫ ∞u2n

w(m−1)/2e−wdw

]= sign(γ)

2m/2−1

π1/2

∫ u2n∨u

2n

u2n∧u2

n

w(m−1)/2e−wdw,

and so, without loss of generality, we suppose that γ > 0, which implies that un > un. Now, if m = 1, then∫ u2n

u2n

w(m−1)/2e−wdw = e−u2n − e−u

2n = e−(B2

n+γ2n)/2σ2

n

(eBnγ/σ

2

− e−Bnγ/σ2)∼ π1/2 Bn

2−3/2

γ

σ2φ

(Bnσn

).

For general m, we can proceed by induction to show that∫ u2n

u2n

w(m−1)/2e−wdw ∼ π1/2 Bmn2m/2−2σm−1

n

γ

σ2φ

(Bnσn

). (C.6)

Indeed, suppose that (C.6) holds for m = M − 2 for an odd positive M . Now, for m = M , integration by parts yields∫ u2n

u2n

wM−1

2 e−wdw =e−(B2

n+γ2n)/2σ2

n

2M−1

2 σM−1n

[eBnγ

σ2 (Bn − γn)M−1 − e−Bnγ

σ2 (Bn + γn)M−1]

+M − 1

2

∫ u2n

u2n

w(M−3)/2e−wdw.

The expression in the square brackets above is such that

eBnγ/σ2

(Bn − γn)M−1 − e−Bnγ/σ2

(Bn + γn)M−1 = 2BMn γ

σ2+ o(BMn ), (n→∞).

Therefore, in light of the induction step, as n→∞,∫ u2n

u2n

w(M−1)/2e−wdw = π1/2 BMn2M/2−2σM−1

n

γ

σ2φ

(Bnσn

)+O

(BM−2n

σM−3n

φ

(Bnσn

)), (C.7)

which proves (C.6) for m = M . From (C.6), we deduce that for m ∈ 1, 3, 5, . . .

G(m)n (B) ∼ 2

Bmnσm−1n

γ

σ2φ

(Bnσn

). (C.8)

Finally, from (C.4), (C.5), and (C.8), we conclude the validity of (C.3). In the case that k is odd and γ = 0, from (C.4)

and the symmetry of the integral term, it is clear that LIn[B](k) = σknE[Zk1[|σnZ|>Bn]

]= 0.

31

Lemma C.2. Let f be of the mixture form given in (2.3) such that f+ and f− are bounded and continuous at 0. Then,

for any threshold sequence B := (Bn)n such that Bn → 0, and any non-negative integer k, as n→∞,

LII,Absn [B](k) ∼ hnλ2Bk+1

n

k + 1C0(f).

Furthermore, if k is odd and limn→∞Bn/h1/2n =∞, then

LIIn [B](k) ∼ hnλBk+1n

k + 1C(f).

Proof. The results follows directly from conditioning on how many jumps occur over the time interval (hni, hn(i+ 1)]

and applying the results of Lemma C.3 below.

Lemma C.3. Let f be a bounded density function, let B = (Bn)n be a positive sequence such that Bn → 0, as n→∞,

and k = 0, 1, . . . . Then, the following assertions hold:

1. For arbitrary constants σ ∈ R+ and γ ∈ R,

E[|γ + σZ + ζ1|k 1[|γ+σZ+ζ1|≤Bn]

]≤ 2‖f‖∞

Bk+1n

k + 1. (C.9)

2. If, additionally, f has the mixture form given in (2.3) such that f+ : [0,∞)→ [0,∞) and f− : (−∞, 0]→ [0,∞)

are left- and right-continuous at 0, respectively, then, as n→∞,

limn→∞

k + 1

Bk+1n

E[|γn + σnZ + ζ1|k 1[|γn+σnZ+ζ1|≤Bn]

]= pf+(0) + qf−(0), (C.10)

where σn = σ√hn and γn = γhn. Furthermore, if k is odd, and Bnh

−1/2n →∞, then

limn→∞

k + 1

Bk+1n

E[(γn + σnZ + ζ1)

k1[|γn+σnZ+ζ1|≤Bn]

]= pf+(0)− qf−(0). (C.11)

Proof of Lemma C.3. For the first assertion, note that

E[|γ + σZ + ζ1|k 1[|γ+σZ+ζ1|≤Bn]

]=

∫R

∫ Bn

−Bn|w|kσ−1φ

(w − x− γ

σ

)dwf(x)dx

≤ ‖f‖∞∫ Bn

−Bn|w|k

∫Rσ−1φ

(w − x− γ

σ

)dxdw

= ‖f‖∞∫ Bn

−Bn|w|kdw,

which implies (C.9). To show the second assertion, let us start by introducing some notation needed in the sequel. Let

φn(z) := φ(z/σn)/σn denote the density of σWhn and, for a, b ∈ −1, 1, let

Ia,bn (w) :=

∫ ∞0

(φn(w − bx− γn) + aφn(−w − bx− γn)) dx.

Note that, for w > 0,

I1,bn (w) = 1− b · sgn(γ)

∫ (w+|γn|)/σn

(w−|γn|)/σnφ(z)dz, I−1,b

n (w) = b

∫ (w+γn)/σn

(−w+γn)/σn

φ(z)dz. (C.12)

32

Without loss of generality, we assume hereafter that γ > 0 and, for future reference, let us also remark that, by the

mean value theorem, I1,bn (w) = 1− 2γbσ−1h

1/2n φ(ζbn(w)), for some ζbn(w) ∈ R.

Let us denote the expectations appearing in (C.10) and (C.11) by Cn,k and Dn,k, respectively, and note that

Cn,k =

∫R

∫R|z + x+ γn|k1[|z+x+γn|≤Bn]f(x)φn(z)dxdz = T

(1)n,k + T

(−1)n,k ,

Dn,k =

∫R

∫R

(z + x+ γn)k1[|z+x+γn|≤Bn]f(x)φn(z)dxdz = T(1)n,k + (−1)kT

(−1)n,k ,

where, for b ∈ −1, 1,

T(b)n,k :=

∫ Bn

0

∫Rf(x)φn(bw − x− γn)dxwkdw =

∫ Bn

0

∫R

[pf+(x)1[x≥0] + qf−(x)1[x<0]

]φn(bw − x− γn)dxwkdw.

Let us analyze the two expressions Cn,k and Dn,k via the generalized sum

H(a)n,k :=

T(1)n,k + aT

(−1)n,k

mn, a ∈ −1, 1,

where mn :=∫ Bn

0wkdw = Bk+1

n /(k + 1). To this end, consider the decomposition

H(a)n,k =

p

mn

∫ Bn

0

∫ ∞0

[f+(x)− f+(0)] [φn(w − x− γn) + aφn(−w − x− γn] dxwkdw

+q

mn

∫ Bn

0

∫ 0

−∞[f−(x)− f−(0)] [φn(w − x− γn) + aφn(−w − x− γn)] dxwkdw

+p

mnf+(0)

∫ Bn

0

Ia,1n (w)wkdw +q

mnf−(0)

∫ Bn

0

Ia,−1n (w)wkdw

=: C(a)

n,k + C(a)n,k + C

(a)n,k. (C.13)

Depending on the sign of a, the last term on the right-hand side of (C.13) equals

C(a)n,k =

pf+(0) + qf−(0) + γO(h

1/2n ), if a = 1,

pf+(0)−qf−(0)mn

∫ Bn0

∫ w+γnσn

(−w+γn)σn

φ(v)dvwkdw, if a = −1.

(C.14)

In order to show the asymptotic behavior of C(−1)n,k , we next show that

Jn,k :=1

mn

∫ Bn

0

∫ w+γnσn

−w+γnσn

φ(v)dvwkdwn→∞−→ 1. (C.15)

First note that, by a change of variables,

Jn,k =σk+1n

mn

∫ Bnσn

0

∫ u+ γnσn

−u+ γnσn

φ(v)dvukdu.

Let u0 > 0 be large enough that∫

(−u0,u0)cφ(v)dv < ε. Let N large enough that Bn/σn > u0 + 1 and |γn/σn| < 1 for

33

all n ≥ N . Then, for n ≥ N ,

0 ≤ 1− Jn,k =σk+1n

mn

∫ Bnσn

0

∫(−u+ γn

σn,u+ γn

σn)cφ(v)dvukdu

≤ σk+1n

mn

∫ u0+1

0

ukdu+σk+1n

mn

∫ Bnσn

u0+1

∫(−u0,u0)c

φ(v)dvukdu

≤ σk+1n

mn

∫ u0+1

0

ukdu+ εσk+1n

mn

∫ Bnσn

0

ukdun→∞−→ ε,

because σk+1n /mn → 0 and both u+ γn/σn > u0 and −u+ γn/σn < −u0 whenever u > u0 + 1. Since ε is arbitrary, we

conclude the first limit in (C.15). Together (C.14) and (C.15) imply that

C(a)n,k

n→∞−→

pf+(0) + qf−(0); if a = 1,

pf+(0)− qf−(0); if a = −1.

(C.16)

We will now show that the first two terms appearing on the right-hand side of (C.13) are such that

limn→∞

C(a)

n,k = limn→∞

C(a)n,k = 0. (C.17)

Given ε > 0, by the continuity of f+ at 0, there exists a δ > 0 such that |f+(x)− f+(0)| < ε/2, for all x ∈ (0, δ). The

ensuing upper bound follows:

|C(a)

n,k| ≤p

mn

∫ Bn

0

∫ δ

0

|f+(x)− f+(0)| [φn(w − x− γn) + φn(w + x+ γn)] dxwkdw

+p

mn

∫ Bn

0

∫ ∞δ

f+(x) [φn(w − x− γn) + φn(w + x+ γn)] dxwkdw

+ f+(0)p

mn

∫ Bn

0

∫ ∞δ

[φn(w − x− γn) + φn(w + x+ γn)] dxwkdw.

Let us denote each of the three terms in the right-hand side of the last inequality by Cn,k,1,Cn,k,2, and Cn,k,3,

respectively. Clearly,

Cn,k,1 ≤ε

2

p

mn

∫ Bn

0

∫ δ

0

[φn(w − x− γn) + φn(−w − x− γn)] dxwkdw ≤ ε,

Cn,k,2 ≤ supw∈(0,Bn)

x∈(δ,∞)

φn(w − x− γn) + supw∈(0,Bn)

x∈(δ,∞)

φn(w + x+ γn).

Note that, for n large enough, min|w − x− γn|, |w + x+ γn| > δ/2 for all w ∈ (0, Bn) and x ∈ (δ,∞). Thus,

Cn,k,2 ≤ 2 sup|z|>δ/2

φn(z)→ 0, (n→∞).

Similarly,

Cn,k,3 ≤ f+(0)2p

mn

∫ Bn

0

∫|z|≥δ/2

φn(z)dzwkdw ≤ 2f+(0)

∫|z|≥δ/2

φn(z)dz → 0, (n→∞).

Therefore,

lim supn→∞

∣∣∣C(a)

n,k

∣∣∣ ≤ lim supn→∞

Cn,k,1 + lim supn→∞

Cn,k,2 + lim supn→∞

Cn,k,3 ≤ ε, (C.18)

34

and, since ε is arbitrary, we conclude that limn→∞ C(a)

n,k = 0. One can similarly show that limn→∞ C(a)n,k = 0. Together

(C.13), (C.16), and (C.17) yield (C.10) and (C.11).

Lemma C.4. Let f be of the mixture form given in (2.3) such that f+ : [0,∞)→ [0,∞) and f− : (−∞, 0]→ [0,∞) are

bounded. Then, for any nonnegative sequence (Bn)n such that Bn → 0, any nonnegative integers k, and any m ≥ 2,

we have:

limn→0

k + 1

Bk+1n

E[|γn + σ∆n

iW + ζ1 + · · ·+ ζm|k 1[|γn+σ∆niW+ζ1+···+ζm|≤Bn]

]= 2f∗m(0), (C.19)

limn→0

k + 1

Bk+1n

E[(γn + σ∆n

iW + ζ1 + · · ·+ ζm)k1[|γn+σ∆n

iW+ζ1+···+ζm|≤Bn]

]= f∗m(0)(1 + (−1)k), (C.20)

as n→∞. Furthermore, if k is odd, f∗m ∈ C1b , and Bn/h

1/2n →∞, then

limn→∞

k + 2

Bk+2n

E[(γn + σ∆n

iW + ζ1 + · · ·+ ζm)k1[|γn+σ∆n

iW+ζ1+···+ζm|≤Bn]

]= 2(f∗m)′(0). (C.21)

Proof of Lemma C.4. The first two relationships (C.19) and (C.20) follow from Lemma C.3 since ζ1 + · · · + ζm

can be seen as a random variable with a density satisfying the conditions of the Lemma. Indeed, for m ≥ 2, the

density of ζ1 + · · · + ζm is the m-fold convolution f∗m, which is actually continuous everywhere under the condition

f ∈ L1(R) ∩ L∞(R). So, we only need to show the second assertion and, as with (C.19) and (C.20), it suffices to

consider m = 1. Let us recall from the proof of Lemma C.3 that the expectation appearing in (C.21), denoted therein

by Dn,k, can be expressed as

Dn,k =

∫R

∫R

(z + x+ γn)k1[|z+x+γn|≤Bn]f(x)φn(z)dxdz = T(1)n,k − T

(−1)n,k ,

where, for b ∈ −1, 1,

T(b)n,k :=

∫ Bn

0

∫Rf(x)φn(bu− x− γn)dxukdu.

Next, changing variables into y = x+ γn for T(1)n,k and into y = −x− γn for T

(−1)n,k ,

T (1)n =

∫ Bn

0

∫Rf(y − γn)φn(u− y)dyukdu, T (−1)

n =

∫ Bn

0

∫Rf(−y − γn)φn(−u+ y)dyukdu

Therefore, invoking the symmetry of φn,

Dn,k

Bk+2n

=1

Bk+2n

∫ Bn

0

∫R(f(y − γn)− f(−y − γn))φn(u− y)dxukdu

=1

Bk+2n

∫ Bn

0

∫R

∫ 1

−1

f ′(yβ − γn)dβyφn(u− y)dyukdu.

Denoting a standard Gaussian variable by Z and changing variables first from u to v = u/Bn, then from y to z = Bnv−y,

and finally from β to α = Bnβ, we can write

Dn,k

Bk+2n

=1

Bn

∫ 1

0

∫R

∫ 1

−1

f ′(yβ − γn)dβyφn(Bnv − y)dyvkdv

=

∫ 1

0

E

(1

Bn

∫ Bn

−Bnf ′((u−B−1

n σnZ)α− γn

)dα(u−B−1

n σnZ))

ukdu.

35

By the Dominated Convergence Theorem and the assumption that h1/2n B−1

n → 0, we conclude that limn→∞Dn,k/Bk+2n =

2f ′(0)/(k + 2).

Lemma C.5. Let B := (Bn)n≥1 be an arbitrary threshold sequence and k = 0, 1, . . . . Then, limn→∞Bn/h1/2n = +∞

if and only if

limn→∞

GIn[B](k)

hk/2n

=σk2k/2

π1/2Γ

(k + 1

2

). (C.22)

Furthermore, when limn→∞Bn/h1/2n = +∞, we have, as n→∞,

GIn[B](k)

hk/2n

−σk2k/2Γ(k+1

2 )

π1/2∼

An,k, if k is even or γ = 0,

An,k + γk+1

σ h(k+1)/2n

(2π

)1/2∑kj=0

(kj

) (−1)k−j

k−j+1 , if k is odd and γ > 0,

An,k − γk+1

σ h(k+1)/2n

(2π

)1/2∑kj=0

(kj

)1

k−j+1 , if k is odd and γ < 0,

(C.23)

where

An,k := −2σφ

(Bnσn

)(Bn

h1/2n

)k−1

+hnπ1/2

σk−22k/2−1

[−2λσ2Γ

(k + 1

2

)+ γ2

(k

2

)Γ

(k − 1

2

)].

Proof. Let us first assume that lim infn→∞Bn/√hn =: L ∈ [0,∞) and let njj≥1 be a subsequence such that

limj→∞Bnj/h1/2nj = L. Then, by the dominated convergence theorem,

GInj [B](k)

hk/2nj

= e−λnE

|γh1/2nj + σZ|k1

[|γh1/2nj

+σZ|≤Bnj

h1/2nj

]

j→∞−→ E[|σZ|k1[|σZ|≤L]

]. (C.24)

The limit value above can be expressed in terms of the lower incomplete gamma function, Γ(s, x) :=∫ x

0us−1e−udu, as

follows:

E[|σZ|k1[|σZ|≤L]

]= 1L 6=0

σk2k/2√π

Γ

(k + 1

2,L2

2σ2

).

This proves that, for the validity of (C.22), it is necessary that limn→∞Bn/h1/2n = +∞. Hereafter, we assume the

later condition. In that case, using again dominated convergence, GIn[B](k)/hk/2n → E

[|σZ|k

], and (C.22) follows from

the well-known formula for the centered moments of a normal r.v. To show the second assertion, we assume that γ > 0

(the cases γ < 0 and γ = 0 are proved similarly). Let us begin by noting that

Dn,k :=GIn[B](k)

hk/2n

−σk2k/2Γ(k+1

2 )

π1/2= e−λnh−k/2n

∫ (Bn−γn)/σn

−(Bn+γn)/σn

|γn + σnz|kφ(z)dz −σk2k/2Γ(k+1

2 )

π1/2. (C.25)

Since −(Bn + γn)/σn) < −γn/σn < (Bn − γn)/σn, we may further decompose the integral term appearing on the

right-hand side above as follows:∫ (Bn−γn)/σn

−(Bn+γn)/σn

|γn + σnz|kφ(z)dz =

∫ (Bn−γn)/σn

−γn/σn(γn + σnz)

kφ(z)dz + (−1)k

∫ −γn/σn−(Bn+γn)/σn

(γn + σnz)kφ(z)dz

=

k∑j=0

(k

j

)γjσk−jh(k+j)/2

n

∫ (Bn−γn)/σn

−γn/σnzk−jφ(z)dz

+ (−1)kk∑j=0

(k

j

)γjσk−jh(k+j)/2

n

∫ −γn/σn−(Bn+γn)/σn

zk−jφ(z)dz. (C.26)

36

Next, by a change of variable,

Dn,k =σk2k/2−1

π1/2

[e−λnΓ

(k + 1

2,

(Bn + γn)2

2σ2n

)− Γ

(k + 1

2

)](C.27)

+σk2k/2−1

π1/2

[e−λnΓ

(k + 1

2,

(Bn − γn)2

2σ2n

)− Γ

(k + 1

2

)](C.28)

+ e−λnσk2k/2−1

π1/2Γ

(k + 1

2,γ2hn2σ2

)[(−1)k − 1

](C.29)

+

k∑j=1

(k

j

)γjσk−jhj/2n

2(k−j)/2−1

π1/2

[Γ

(k − j + 1

2,

(Bn − γn)2

2σ2n

)+ (−1)2k−jΓ

(k − j + 1

2,

(Bn + γn)2

2σ2n

)

+ Γ

(k − j + 1

2,γ2hn2σ2

)(−1)k−j [1− (−1)k]

]. (C.30)

We denote the previous four terms P(1,+)n , P

(1,−)n , P

(2)n , and P

(3)n . From here we can use the asymptotic properties of

the lower and upper incomplete gamma functions in order to analyze the previous four terms on the right-hand side

of (C.30). Indeed, for the first two terms,

P (1,±)n = −σ

k2k/2

π1/2Γ

(k + 1

2,

(Bn ± γn)2

2σ2n

)− λhn

σk2k/2−1

π1/2Γ

(k + 1

2,

(Bn ± γn)2

2σ2n

)+ o(hn)

=− σφ(Bnσn

)(Bn

h1/2n

)k−1

− λhnσk2k/2−1

π1/2Γ

(k + 1

2

)+ h.o.t., (n→∞), (C.31)

where Γ(s, x) :=∫∞xus−1e−udu denotes the upper incomplete gamma function and we have used the asymptotics

Γ(s, x) ∼ xs−1e−x as x→∞. For the second term, note that P(2)n = 0 is k if even, while, for odd k,

P (2)n ∼ −γ

k+1

σ

h(k+1)/2n

k + 1

(2

π1/2

)1/2

. (C.32)

Lastly, for the last term in line (C.30), we have, as n→∞,

P (3)n ∼

hn(k2

)γ2σk−2 2k/2−1

π1/2 Γ(k−1

2

); when k is even,

hn(k2

)γ2σk−2 2k/2−1

π1/2 Γ(k−1

2

)+ γk+1

σ h(k+1)/2n

(2π

)1/2∑kj=1

(kj

) (−1)k−j

k−j+1 ; when k is odd.

(C.33)

Indeed, for odd k, we make use of the fact that Γ(s, x) ∼ xs/s as x→ 0 to conclude that

γjσk−jhj/2n

2(k−j)/2−1

π1/2Γ

(k − j + 1

2,γ2hn2σ2

)∼ γk+1

2√

2πσ2

h(k+1)/2n

k − j + 1, (n→∞),

for each j = 1, 2, · · · , k. For even k, using that Γ(x, s) = Γ(s) − Γ(s, x) and the asymptotics Γ(s, x) ∼ xs−1e−x, as

x→∞, the term for odd j can be proved to be of order

hj− k−1

2n Bk−jn φ

(Bnσn

),

which is negligible compared to the first term in (C.31). The term in (C.33) corresponds to j = 2, which is the

dominant among those terms with even j. This concludes the proof.

37

Lemma C.6. Let f be of the mixture form given in (2.3) such that f+ : [0,∞) → [0,∞) and f− : (−∞, 0] → [0,∞)

are bounded functions and let (Bn)n≥1 be a sequence such that limn→∞Bn/h1/2n =∞. Then, as n→∞,

E[(γn + σnZ)21[|∆n

i X|>Bn,∆ni N 6=0]

]∼ λσ2h2

n, (C.34)

E[(γn + σnZ)1[|∆n

i X|>Bn,∆ni N 6=0]

]= γλh2

n + o(h2n) + o(Bnh

3/2n ). (C.35)

Proof. To show (C.34), note that, for any nonnegative integer k,

E[|Z|k1[|∆n

i X|>Bn,∆ni N 6=0]

]= e−λn

∞∑j=1

λjnj!

E[|Z|k1[|γn+σnZ+ζ1+···+ζj |>Bn]

]∼ λhnE

[|Z|k

], (n→∞),

since, by dominated convergence theorem, E[|Z|k1[|γn+σnZ+ζ1+···+ζj |>Bn]

]→ E

[|Z|k

], as n→∞. It is now clear that,

as n→∞,

E[(γn + σnZ)21[|∆n

i X|>Bn,∆ni N 6=0]

]= γ2λh3

n + 2λσh3/2n O(hn) + σ2λh2

n ∼ σ2λh2n.

Let us denote the left-hand side of (C.35) by An, and then note that

An = γne−λn

∞∑j=1

λjnj!

P (|γn + σnZ + ζ1 + · · ·+ ζj | > Bn) + σne−λn

∞∑j=1

λjnj!

E[Z1[|γn+σnZ+ζ1+···+ζj |>Bn]

]= γλh2

n + o(h2n) + Tn +Rn.

where

Tn := −σλe−λnh3/2n E

[Z1[|γn+σnZ+ζ1|≤Bn]

], Rn := σne

−λn∞∑j=2

λjnj!

E[Z1[|γn+σnZ+ζ1+···+ζj |≤Bn]

].

Now, take n ∈ N large enough such that −Bn < γn < Bn and consider the following decomposition:

E[Z1[|γn+σnZ+ζ1|≤Bn]

]=

∫ ∞(Bn−γn)/σn

∫ Bn−γn−σnz

−(Bn+γn+σnz)

qf−(x)dxzφ(z)dz

+

∫ −(Bn+γn)/σn

−∞

∫ Bn−γn−σnz

−(Bn+γn+σnz)

pf+(x)dxzφ(z)dz

+

∫ (Bn−γn)/σn

−(Bn−γn)/σn

[∫ 0

−(Bn+γn+σnz)

qf−(x)dx+

∫ (Bn+γn+σnz)

0

pf+(x)dx

]zφ(z)dz

=: H(1)n +H(2)

n +H(3)n .

In light of the uniform bounds on f+ and f−, we have

|H(1)n |

2Bn≤ q‖f−‖∞

∫ ∞(Bn−γn)/σn

zφ(z)dz = q‖f−‖∞φ(Bn − γnσn

), (C.36)

|H(2)n |

2Bn≤ p‖f+‖∞

∫ −(Bn−γn)/σn

−∞zφ(z)dz = p‖f+‖∞φ

(Bn + γnσn

). (C.37)

For H(3)n , an application of the dominate convergence theorem yields

limn→∞

H(3)n

Bn= [pf+(0) + qf−(0)]

∫ ∞−∞

zφ(z)dz = 0. (C.38)

38

From (C.36)-(C.38), it is clear that Tn = o(Bnh3/2n ), as n→∞. On the other hand, the analysis used above also shows

that, for any j ∈ N, we have E[Z1[|γn+σnZ+ζ1+···+ζj |≤Bn]

]= o(Bn) and, thus, Rn = o(Bnh

5/2n ).

Lemma C.7. Given γ 6= 0, σ > 0, h > 0, and r > 0, the quantity given by

Dh,r := E[∣∣∣γh+ σh1/2Z

∣∣∣r]− E[∣∣∣σh1/2Z

∣∣∣r] ,is such that Dh,r ∼ rσr−2hr/2+1γ2 2r/2−1

π1/2 Γ(r+1

2

), as h→ 0+.

Proof. From Winkelbauer (2012), for any r > −1,

E[∣∣∣γh+ σh1/2Z

∣∣∣r] = σrhr/22r/2

π1/2Γ

(r + 1

2

)1F1

(−r

2,

1

2,−γ

2h

2σ2

),

where 1F1 (a, b, z) denotes Kummer’s confluent hypergeometric function, which is defined by the expansion∑∞i=0

a(i)

b(i)zi

i!

and a(i) := a(a+ 1) · · · (a+ i− 1) and a(0) = 1. The stated asymptotics follow directly from here.

Lemma C.8. Let r ∈ [0,∞) and suppose that E[|ζ1|r] <∞. Then, as n→∞,

E[|γn + σ∆n

1W + ∆n1J |

rI[∆n

1N 6=0]

]= λhnE[|ζ1|r] + o(hn). (C.39)

Proof. Let Z be a standard normal variable. For future reference, note that E[|ζ1|r] <∞ implies that E[|ζ1+· · ·+ζm|r] <

∞, for all m ≥ 1, since |ζ1 + · · ·+ ζm|r ≤ mr maxi≤m|ζi|r ≤ mr(|ζ1|r + · · ·+ |ζm|r). Now, fix m ∈ N and let

arn(m) := E [|γn + σ∆n1W + (ζ1 + · · ·+ ζm)|r] e−λn λ

mn

m!.

Clearly, for n large enough so that hn ≤ 1, |γn + σ∆n1W + (ζ1 + · · ·+ ζm)|r ≤ 3r(|γ|r +σ|Z|r + |ζ1 + · · ·+ ζm|r), which

has finite expectation. Thus, from the dominated convergence theorem,

limn→∞

h−mn arn(m) =λm

m!E [|ζ1 + · · ·+ ζm|r] . (C.40)

Finally, by conditioning on the total number of jumps over the time interval [(i− 1)hn, ihn),

E[|γn + σ∆n

iW + ∆ni J |

rI[∆n

i N 6=0]

]=∑m≥1

E [|γn + σ∆niW + ζ1 + · · ·+ ζm|r]

e−λn

m!λmn = arn(1) + o(hn),

from which the results directly follows.

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