STOCHASTIC COMPARISONS OF FUZZY STOCHASTIC PROCESSESshodhganga.inflibnet.ac.in/bitstream/10603/5336/13/13... · 2015-12-04 · stochastic comparison of random processes with applications

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,


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,


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


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-


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.


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)


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.


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.


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


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


≥

∞∫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

)


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].


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]


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


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


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)


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)


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).


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).


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


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.


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


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


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.


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).

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STOCHASTIC COMPARISONS OF FUZZY STOCHASTIC PROCESSESshodhganga.inflibnet.ac.in/bitstream/10603/5336/13/13... · 2015-12-04 · stochastic comparison of random processes with applications