STOCHASTIC COMPARISONS OF FUZZY STOCHASTIC...
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CHAPTER
FIVE
STOCHASTIC COMPARISONS OF FUZZY
STOCHASTIC PROCESSES
Abstract
Chapter 5 investigates the stochastic comparison of fuzzy stochastic
processes. This chapter introduce the concept of stochastic compar-
ison of fuzzy stochastic processes. The condition that manifests the
stochastic inequality is realized in terms of an increasing functional
f . Chapter 5 ends with the concluding section. This section of con-
clusion includes the summary of the results of this thesis.
The contents of this chapter form the substance of the paper
entitled ”Stochastic comparisons of fuzzy stochastic processes”,
accepted for publication in the International Journal Reflection
des ERA-Journal of Mathematical Sciences, India.
111
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 112
5.1 Introduction
The theory of fuzzy random variables is a natural extension
of classical real valued random variables or random vectors.
Fuzzy random variables have many special properties. This
allows new meanings for the classical probability theory. As
a result of advancement in this area in the past three decades
the theory of fuzzy random variables with diverse applications
has become one of new and active branches in probability the-
ory. In reality we often come across with random experiments
whose outcomes are not numbers but are expressed in inexact
linguistic terms, which varies with time t. Such linguistic terms
will be represented by a dynamic fuzzy set [49]. This is a typ-
ical fuzzy stochastic phenomenon with prolonged time. Fuzzy
random variables [33, 34, 44, 67] are mathematical characteriza-
tions for fuzzy stochastic phenomena, but only one point of time
description. For the formulation of a fuzzy stochastic process,
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 113
fuzzy random variables should be considered repeatedly and
even continuously to describe and investigate the structure of
their family. As a desideratum, the study of fuzzy stochastic
processes is essential.
Kwakernaak [33, 34] introduced the notion of a fuzzy ran-
dom variable as a measurable functions F : Ω → F(R), where
(Ω,A ,P) is a probability space and F(R) denotes all piecewise
continuous functions u : R → [0, 1]. Puri and Ralescu [44]
defined the concept of a fuzzy random variable as a function
F : Ω → F(Rn) where (Ω,A ,P) is a probability space and F(Rn)
denotes all functions u : Rn→ [0, 1] such that x ∈ Rn; u(x) ≥ α
is a non-empty and compact for each α ∈ (0, 1]. In this chapter, a
concept of fuzzy random variable, slightly different than that of
Kwakernaak [33, 34] and Puri [44] is introduced. It is defined as
a measurable fuzzy set valued function X : Ω→ F0(R), where R
is the real line, (Ω,A ,P) is a probability space,
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 114
F0(R) = A : R→ [0, 1] and x ∈ R; A(x) ≥ α is a bounded closed
interval for each α ∈ (0, 1]. Guangyuan Wang et.al., [18] have in-
troduced the general theory of fuzzy stochastic processes, which
include the definitions of fuzzy random function, fuzzy stochas-
tic processes. Earnest Lazarus Piriyakuar et.al., [12] have stud-
ied various stochastic comparison of fuzzy random variables.
In this chapter the concept of stochastic comparison is extended
to fuzzy stochastic processes. Congruous to stochastic compar-
isons of classical random variables, stochastic comparisons for
functionals of fuzzy stochastic processes, which are of practical
importance are derived.
The stochastic comparison of two fuzzy random variables
whose end points of each α-cut is univariate in nature can be
generalized and the resulting stochastic comparison is nothing
but the stochastic comparison of two fuzzy stochastic processes.
In many applied problems the exact calculation of quantities of
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 115
interest which are obscured by perceptional deficiencies the re-
sulting fuzzy stochastic process pose a variety of complexities.
In such cases the only remedy is to compute bounds on these
parameters by comparing the given fuzzy stochastic process
with a simpler fuzzy stochastic process. This kind of stochastic
comparison has great relevance in reliability problems. In this
chapter, stronger type of comparison of two fuzzy stochastic pro-
cesses is introduced. Gordon Pledger et. al [15] have discussed
stochastic comparison of random processes with applications in
reliability.
In Section 5.2, some results related to dynamic fuzzy sets,
fuzzy random variables, fuzzy random vectors, fuzzy random
function and fuzzy stochastic processes are introduced.
In Section 5.3, the concept of stochastic comparison of fuzzy
stochastic processes is introduced. Conditions are obtained un-
der which the fuzzy stochastic process X(t); t ≥ 0 stochas-
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 116
tically larger than its counter part Y(t); t ≥ 0 implies that
f X(t); t ≥ 0 ≥st f Y(t); t ≥ 0 for increasing functional f , and
other properties of stochastic comparison are introduced.
5.2 Preliminaries
Let R be the real line and (R,B) be the Borel measurable space.
Let F0(R) denote the set of fuzzy subsets A : R→ [0, 1] with the
following properties:
1. x ∈ R; A(x) = 1 , φ.
2. Aα = x ∈ R; A(x) ≥ α is a bounded closed interval in R for
each α ∈ (0, 1]. i.e., Aα =[(Aα)L , (Aα)U
]where
(Aα)L = inf Aα and (Aα)U = sup Aα
(Aα)L, (Aα)U∈ Aα, −∞ < (Aα)L and (Aα)U < ∞ for each
α ∈ (0, 1]. A ∈ F0(R) is called a bounded closed fuzzy
number.
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 117
Definition 5.2.1 ([49]). A(t), t ∈ T ⊂ R is known as a dynamic
fuzzy set in U where U is a non empty set with respect to T if
A(t) ∈ F(U), the set of all fuzzy subsets of U, for each t ∈ T. In
particular A(t); t ∈ T is called a normal dynamic fuzzy set if
A(t) ∈ F0(R) for each t ∈ T.
Definition 5.2.2. Let A(t) be a normal dynamic fuzzy set with
respect to T and I(R) = [x, y]; x, y ∈ R, x ≤ y.
Let Aα : T→ I(R) defined as
t 7→ Aα(t) =(A(t)
)α
=[(Aα)L (t), (Aα)U (t)
].
Then Aα(t) is known as the level function of A(t). Aα is an
interval valued mapping on T.
Definition 5.2.3 ([18]). Let (Ω,A ,P) be a probability space. A
fuzzy set valued mapping X : Ω → F0(R) is called a fuzzy
random variable if for each B ∈B and every α ∈ (0, 1],
X−1α (B) = ω ∈ Ω; Xα(ω) ∩ B , φ ∈ A .
A fuzzy set valued mapping X : Ω→ Fm0 (R) = F0(R)× · · · ×F0(R)
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 118
represented by X(ω) =(X(1, ω),X(2, ω), . . . ,X(m, ω)
)is known
as a fuzzy random vector if for each k, 1 ≤ k ≤ m, X(k, ω) is a
fuzzy random variable.
Definition 5.2.4 ([18]). Let (Ω,A ,P) be a probability space and
X a set valued mapping X : Ω→ I(R) defined as
ω 7→ X(ω) = [XL(ω),XU(ω)]
Then X(ω) = [XL(ω),XU(ω)] is called a random interval if XL(ω)
and XU(ω) are both random variables on (Ω,A ,P).
Theorem 5.2.1 ([18]). X(ω) is a fuzzy random variable if and only
if Xα(ω) =[(Xα)L (ω), (Xα)U (ω)
]is a random interval for each
α ∈ (0, 1] and
X(ω) = ∪α∈(0,1]
α Xα(ω) = ∪α∈(0,1]
α[(Xα)L (ω), (Xα)U (ω)
](5.2.1)
This theorem is useful in the construction of various fuzzy
sets made in terms of their corresponding α-cuts. Most of the
results of this chapter are proved for the corresponding α-cuts.
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 119
With the help of the above theorem we can establish the results
for the corresponding fuzzy sets.
A family of fuzzy random variables X(t) = X(t, ω); t ∈ T is
known as a fuzzy random function. The parameter set T can be
viewed as any one of the following: R,R+ = [0,∞), [a, b] ⊂ R,
Z = 0,±1,±2, . . ., Z+ = 0, 1, 2, . . ., 1, 2, . . . ,m and so on. In all
these cases the parameter t ∈ T can be viewed as time.
If T = Z or Z+ then a fuzzy random sequence can be realized. If
T = R or R+ or [a, b], X(t) is known as a fuzzy stochastic process.
Definition 5.2.5 ([18]). A fuzzy random function X(t) = X(t, ω),
t ∈ T is a fuzzy set valued function from the space T×Ω to F0(R).
X(t, ·) is a fuzzy random variable on (Ω,A ,P) for each fixed t ∈ T
and X(·, ω) is a normal Dynamic fuzzy set with respect to the
parameter set T, for each fixed ω ∈ Ω. X(·, ω) is called a fuzzy
sample function or a fuzzy trajectory.
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 120
Definition 5.2.6 ([18]). Let (Ω,A ,P) be a probability space,
T ⊂ R and X a set valued mapping.
X : T ×Ω→ I(R) defined as
(t, ω)→ X(t, ω) = [XL(t, ω),XU(t, ω)]
is known as an interval valued random function if XL(t,ω) and
XU(t,ω) are both random functions.
The following theorem is important for the construction of
fuzzy stochastic processes using their corresponding α-cuts. In
this chapter the stochastic comparison of fuzzy stochastic pro-
cesses are proved in terms of their corresponding α-cuts. With
the aid of the following theorem stochastic comparison of fuzzy
stochastic processes can be realized.
Theorem 5.2.2 ([18]). X(t) = X(t, ω); t ∈ T is a fuzzy random
function if and only if for each α ∈ (0, 1].
Xα(t) = Xα(t, ω), t ∈ T =[
(Xα)L (t, ω), (Xα)U (t, ω)], t ∈ T
is
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 121
an interval random function for each t ∈ T. and every ω ∈ Ω and
X(t, ω) = ∪α∈(0,1]
αXα(t, ω) = ∪α∈(0,1]
α[(Xα)L (t, ω), (Xα)U (t, ω)
](5.2.2)
Definition 5.2.7. The fuzzy random variable X is stochastically
larger than the fuzzy random variable Y if
P((Xα)L > a
)∨ P
((Xα)U > a
)≥
P((Yα)L > a
)∨ P
((Yα)U > a
)symbolically it is denoted as X ≥st Y.
5.3 Stochastic comparison of fuzzy stochastic
processes
Theorem 5.3.1. Let X and Y be fuzzy random variables.
If X ≥st Y then E[X] ≥ E[Y].
Proof. Assume first that X and Y are non-negative fuzzy ran-
dom variables. Then for α ∈ (0, 1],
E[(Xα)L
]∨ E
[(Xα)U
]=
∞∫0
P((Xα)L > a
)da ∨
∞∫0
P((Xα)U > a
)da
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 122
≥
∞∫0
P((Yα)L > a
)da ∨
∞∫0
P((Yα)U > a
)da
= E[(Yα)L
]∨ E
[(Yα)U
]Generally one can express any fuzzy random variable Z as the
difference of two non-negative fuzzy random variables.
Let Z = Z+− Z−.
i.e., For each x ∈ R,
Z+(ω)(x) =
Z(ω)(x); Z(ω)(x) ≥ 0
0; Z(ω)(x) < 0
Z−(ω)(x) =
0; Z(ω)(x) ≥ 0
−Z(ω)(x); Z(ω)(x) < 0
X ≥st Y implies
P(Xα)L > a
∨ P
(Xα)U > a
≥ P
(Yα)L > a
∨ P
(Yα)U > a
Let X(ω)(x) ≥ 0 and Y(ω)(x) ≥ 0. Then
P((
Xα+)L > a
)∨ P
((Xα
+)U > a)≥ P
((Yα+)L > a
)∨ P
((Yα+)U > a
)
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 123
This shows that X+≥
st Y+.
Similarly one can prove X− ≥st Y−.
The proof of other cases are similar.
∴ E[(Xα)L
]∨ E
[(Xα)U
]= E
[(Xα
+)L]∨ E
[(Xα
+)U]
−
[E(Xα−)L]∨
[E(Xα−)U
]≥ E
[(Yα+)L]
∨ E[(
Yα+)U]−
[E(Yα−
)L]∨
[E(Yα−
)U]= E
((Yα)L
)∨ E
((Yα)U
)The above inequality is true for each α ∈ (0, 1].
∴ E(∪
α∈(0,1]α (Xα)L
)∨ E
(∪
α∈(0,1]α (Xα)U
)≥ E
(∪
α∈(0,1]α (Yα)L
)∨ E
(∪
α∈(0,1]α (Yα)U
)
This shows that
E[∪
α∈(0,1]α[(Xα)L , (Xα)U
]]≥ E
[∪
α∈[0,1]α[(Yα)L , (Yα)U
]]
i.e., E[X] ≥ E[Y].
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 124
Theorem 5.3.2. If X and Y are fuzzy random variables then
X ≥st Y if and only if E[ f (X)] ≥ E[ f (Y)] for all increasing func-
tions f .
Proof. Let X ≥st Y, and f be an increasing function.
Let f−1(a) = infx; f (x) ≥ a. Then
P
f (Xα)L > a∨ P
f (Xα)U > a
= P
(Xα)L > f−1(a)
∨ P
(Xα)U > f−1(a)
≥ P
(Yα)L > f−1(a)
∨ P
(Yα)U > f−1(a)
= P
f (Yα)L > a
∨ P
f (Yα)U > a
∴ f (Xα)L
∨ f (Xα)U≥
st f (Yα)L∨ f (Yα)U .
The above inequality is true for each α ∈ (0, 1].
∴ ∪α∈(0,1]
α(
f (Xα)L∨ f (Xα)U
)≥
st∪
α∈(0,1]α(
f (Yα)L∨ f (Yα)U
)This shows that
∪α∈(0,1]
α[
f (Xα)L , f (Xα)U]≥
st∪
α∈(0,1]α[
f (Yα)L , f (Yα)U]
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 125
This shows that f (X) ≥st f (Y).
By Theorem 5.3.1, E[ f (X)] ≥ E[ f (Y)].
Definition 5.3.1. Fuzzy stochastic process X(t); t ≥ 0 is said
to be stochastically larger than the fuzzy stochastic process
Y(t); t ≥ 0 denoted as
X(t); t ≥ 0 ≥stY(t); t ≥ 0
if for α ∈ (0, 1] and for each choice of 0 ≤ t1 < t1 < t2 < · · · < tn;
n = 1, 2, . . . .((Xα)L (t1) · · · (Xα)L (tn)
)≥
st((Yα)L (t1) · · · (Yα)L (tn)
)and(
(Xα)U (t1) · · · (Xα)U (tn))≥
st((Yα)U (t1) · · · (Yα)U (tn)
)(5.3.1)
If we consider continuous increasing funtionals f , then the
stochastic comparison of f(X(t); t ≥ 0
)with f (Y(t); t ≥ 0) is a
particular case of (5.3.1).
For M > 0, let D[0,M] denote the space of all real interval
valued functions on [0,M] whose end points are right
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 126
continuous and have left hand limits. Let SM denote the class of
fuzzy stochastic processes z(t); t ≥ 0 such that
P[z(t) = [zL(t,w), zU(t,w)]; 0 ≤ t ≤M ∈ D(0,M) ⊂ I(R)
]= 1
Let zM denote the fuzzy stochastic processes z(t); 0 ≤ t ≤M.
For n = 1, 2, . . . , we define zMn , the nth approximation of zM
by
(ZMn )L(t) =
(z)L(in−1M) for in−1M ≤ t ≤ (i + 1)n−1M;
0 ≤ i ≤ n − 1
(z)L(M) for t = M
and
(ZMn )U(t) =
(z)U(in−1M) for in−1M ≤ t ≤ (i + 1)n−1M;
0 ≤ i ≤ n − 1
(z)U(M) for t = M
Theorem 5.3.3. Let X and Y be n-dimensional fuzzy random
vectors and X′ and Y′ be n′-dimensional random vectors such
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 127
that X ≥st Y, X′ ≥st Y′ with X and X′ independent and Y and Y′
independent then
(X,X′) ≥st (Y,Y′)
where (X,X′) denotes the (n + n′) dimensional fuzzy random
vector (X1, . . . , Xn, X′1, . . . , X′n).
Proof. Let f (x, x′) be a real valued increasing function of (n+n1)
arguments such that E f (X,X′) and E f (Y,Y′) exist.
Let X∗ be an n′-dimensional fuzzy random vector which is
independent of X and of Y and follows the same distribution as
X′. Then
E f(Xα)L ,
(X′α
)L∣∣∣(X′α)L =
(x′α
)L
= E f((Xα)L ,
(X∗α
)L|(X∗α
)L =(x′α
)L)
≥ E f((Yα)L ,
(X∗α
)L|(X∗α
)L =(x′α
)L)
and
E f((
X∗α)U ,
(X′α
)U|(X′α
)U =(x′α
)U)
= E f((Xα)U ,
(X∗α
)U|(X∗α
)U =(x′α
)U)
≥ E f((Yα)U ,
(X∗α
)U|(X∗α
)U =(x′α
)U)
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 128
where x′ is a fuzzy number.
Since f is increasing in its first n arguments, it follows that
E f((Xα)L ,
(X′α
)L)≥ E f
((Yα)L ,
(X∗α
)L)
and
E f((Xα)U ,
(X′α
)U)≥ E f
((Yα)U ,
(X∗α
)U)
Invoking the same argument, it follows that
E f((Yα)L ,
(X∗α
)L)≥ E f
((Yα)L ,
(Y′α
)L)
and
E f((Yα)U ,
(X∗α
)U)≥ E f
((Yα)U ,
(Y′α
)U)
Combining the above set of inequalities,
E f((Xα)L ,
(X′α
)L)≥ E f
((Yα)L ,
(Y′α
)L)
and
E f((Xα)U ,
(X′α
)U)≥ E f
((Yα)U ,
(Y′α
)U)
The above inequalities are true for each α ∈ (0, 1].
∴ E f(∪
α∈(0,1]α (Xα)L , ∪
α∈(0,1]α(X′α
)L)
≥ E f(∪
α∈(0,1]α (Yα)L , ∪
α∈(0,1]α(Y′α
)L)
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 129
and
E f(∪
α∈(0,1]α (Xα)U , ∪
α∈(0,1]α(X′α
)U)
≥ E f(∪
α∈(0,1]α (Yα)U , ∪
α∈(0,1]α(Y′α
)U)
These two inequalities together imply
E f(∪
α∈(0,1]α[(
(Xα)L ,(X′α
)L),((Xα)U ,
(X′α
)U)])
≥ E f(∪
α∈(0,1]α[(
(Yα)L ,(Y′α
)L),((Yα)U ,
(Y′α
)U)])
This shows that E f (X,X′) ≥ E f (Y,Y′).
Applying Theorem 5.3.2 it follows that
(X,X′) ≥st (Y,Y′)
Theorem 5.3.4. Let M > 0, X(t), t ≥ 0 ∈ SM, Y(t); t ≥ 0 ∈ SM
and Γ ⊂ D[0,M] such that for α ∈ (0, 1] and for n = 1, 2, . . .
P[XMα ∈ Γ, YM
α ∈ Γ, XMn,α ∈ Γ, YM
n,α ∈ Γ] = 1.
Let f : Γ→ (−∞,∞) be continuous and increasing. Then
X(t), t ≥ 0 ≥stY(t), t ≥ 0 implies f (XM) ≥st f (YM).
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 130
Proof. By stipulation for each α ∈ (0, 1],
PXMα ∈ Γ,YM
α ∈ Γ,XMn,α ∈ Γ,YM
n,α ∈ Γ,n = 1, 2, ...
= 1
Then as n→∞ and for each α ∈ (0, 1],
P[(
XMn,α
)L→
(XMα
)L,(YM
n,α
)L→
(YMα
)L]
= 1
and P[(
XMn,α
)U→
(XMα
)U,(YM
n,α
)U→
(YMα
)U]
= 1
Since f is continuous on Γ as n→∞ it follows that
P[
f(XM
n,α
)L→ f
(XMα
)L, f
(YM
n,α
)L→ f
(YMα
)L]
= 1
and P[
f(XM
n,α
)U→ f
(XMα
)U, f
(YM
n,α
)U→ f
(YMα
)U]
= 1
Since X(t); t ≥ 0 ≥stY(t); t ≥ 0 implies for each α ∈ (0, 1],((
XMα
)L(0),
(XMα
)L(M
n
), . . . ,
(XMα
)L(M)
)≥
st((
YMα
)L(0),
(YMα
)L(M
n
), . . . ,
(YMα
)L(M)
)and
((XMα
)U(0),
(XMα
)U(M
n
),(XMα
)U( Mn − 1
), . . . ,
(XMα
)U(M)
)≥
st((
YMα
)U(0),
(YMα
)U(M
n
), . . . ,
(YMα
)U(M)
)Since f is an increasing function over the (n + 1) values
(XMα
)L(0), . . . ,
(XMα
)L(M) and
(XMα
)U(0) . . .
(XMα
)U(M).
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 131
Then by the stochastic comparison of two vectors,
f(XM
n,α
)L≥
st f(YM
n,α
)Land f
(XM
n,α
)U≥
st f(YM
n,α
)U
Then as n→∞
f(XMα
)L≥
st f(YMα
)Land f
(XMα
)U≥
st f(YMα
)U.
Then f(∪
α∈(0,1]α(XMα
)L)≥
st f(∪
α∈(0,1]α(YMα
)L)
and f(∪
α∈(0,1]α(XMα
)U)≥
st f(∪
α∈(0,1]α(YMα
)U)
These two inequalities together imply that
f (XM) ≥st f (YM)
Theorem 5.3.5. For the fuzzy stochastic process X(t),Y(t),X′(t)
and Y′(t) let
X(t); t ≥ 0 ≥stY(t); t ≥ 0
and X′(t); t ≥ 0 ≥stY′(t); t ≥ 0
with X(t); t ≥ 0 independent of X′(t), t ≥ 0 and Y(t), t ≥ 0
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 132
independent of Y′(t), t ≥ 0. Then
X(t) + X′(t), t ≥ 0 ≥stY(t) + Y′(t); t ≥ 0
Proof. Let f be an increasing function on 0 ≤ t1 ≤ · · · ≤ tk. Then
by Theorem 5.3.3.
f[(Xα)L (t1) +
(X′α
)L (t1), . . . , (Xα)L (tk) +(X′α
)L (tk)]
≥st f
[(Yα)L (t1) +
(Y′α
)L (t1), . . . , (Yα)L (tk) +(Y′α
)L (tk)]
f[(Xα)U (t1) +
(X′α
)U (t1), . . . , (Xα)U (tk) +(X′α
)U (tk)]
≥st f
[(Yα)U (t1) +
(Y′α
)U (t1), . . . , (Yα)U (tk) +(Y′α
)U (tk)]
It follows from the definition that
X(t) + X′(t); t ≥ 0 ≥stY(t) + Y′(t); t ≥ 0.
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 133
5.4 Conclusions
The embodiment of this thesis establish the following important
results.
1. Let
fn
andgn
be sequence of canonical positive fuzzy
numbers. Let Xn and Yn be sequence of fuzzy random
variables,
Xn = op
(fn)
denotes Xn is of smaller order in probability
than
fn
and Xn = Op(gn
)denotes Xn is at most of order
gn
in probability then
(i) XnYn = op
(fn gn
),(XnYn = Op
(fn gn
))(ii) |Xn|
S = op
(f sn
),(|Xn|
s = Op
(f sn
))(iii) Xn+Yn=op
(max
(fn, gn
)),(Xn + Yn = Op
(max
(fn, gn
))).
2. (Helly’s theorem) If (i) non-decreasing sequence of fuzzy
probability distribution function Fn(x) converges to the
fuzzy probability distribution function F(x), (ii) the fuzzy
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 134
valued function g(x) is everywhere continuous and (iii) a, b
are continuity points of F(x) then for α ∈ (0, 1]
limn→∞
b∫a
gLα(x)dFmin
n
(x(•)β
)∧ gU
α (x)dFmaxn
(x(•)β
)
=
b∫a
gLα(x)dFmin
(x(•)β
)∧ gU
α (x)dFmax(x(•)β
)3. (Helly Bray Theorem)
(i) The fuzzy valued function g(x) is continuous.
(ii) The fuzzy probability distribution function
Fn(x)→ F(x) in each continuity point of F(x) and
(iii) For any ε > 0 we can find A such that
A∫−∞
∣∣∣gLα(x)
∣∣∣ dFminn
(x(•)β
)∧
∣∣∣gUα (x)
∣∣∣ dFmaxn
(x(•)β
)
+
∞∫A
∣∣∣gLα(x)
∣∣∣ dFminn
(x(•)β
)∧
∣∣∣gUα (x)
∣∣∣ dFmaxn
(x(•)β
)< ε
for all n = 1, 2, 3, . . . , then
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 135
limn→∞
∞∫−∞
gLα(x)dFmin
n
(x(•)β
)∧ gU
α (x)dFmaxn
(x(•)β
)
=
∞∫−∞
gLα(x)dFmin
(x(•)β
)∧ gU
α (x)dFmax(x(•)β
)4. If Y is a fuzzy random variable on (Ω, σ,m), G a fuzzy sub
σ-algebra of σ, then there is always a regular conditional
distribution function for Y given G.
5. If Y be a fuzzy random variable on (Ω, σ,m), G a fuzzy
sub σ-algebra of σ, then there exists a regular conditional
probability for Y given G.
6. Baye’s theorem is valid in the Krzysztof Piasecki’s proba-
bility space defined in terms of fuzzy relation less than.
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 136
7. Let X and Y be n-dimensional fuzzy random vectors X′ and
Y′ be n′-dimensional random vectors such that X ≥st Y,
X′ ≥st Y′ with X and X′ are independent and Y and Y′ are
independent then
(X,X′) ≥st (Y,Y′)
where (X,X′) denotes the (n+n′) dimensional fuzzy random
vector (X1, . . . ,Xn, X′1, . . . ,X′n).