Advanced
RENEWAL AND RENEWAL REWARD THEORIES FOR T-INDEPENDENT FUZZY RANDOM VARIABLES†
RENEWAL AND RENEWAL REWARD THEORIES FOR T-INDEPENDENT FUZZY RANDOM VARIABLES†
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 607-625
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : April 29, 2015
  • Accepted : June 29, 2015
  • Published : September 30, 2015
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
JAE DUCK KIM
DUG HUN HONG

Abstract
Recently, Wang et al. [Computers and Mathematics with Ap-plications 57 (2009) 1232-1248.] and Wang and Watada [Information Sci-ences 179 (2009) 4057-4069.] studied the renewal process and renewal reward process with fuzzy random inter-arrival times and rewards under the T-independence associated with any continuous Archimedean t-norm. But, their main results do not cover the classical theory of the random elementary renewal theorem and random renewal reward theorem when fuzzy random variables degenerate to random variables, and some given assumptions relate to the membership function of the fuzzy variable and the Archimedean t-norm of the results are restrictive. This paper improves the results of Wang and Watada and Wang et al. from a mathematical per-spective. We release some assumptions of the results of Wang and Watada and Wang et al. and completely generalize the classical stochastic renewal theorem and renewal rewards theorem. AMS Mathematics Subject Classification : 60A86.
Keywords
1. Introduction
The theory of fuzzy sets, introduced by Zadeh [27 , 28] , has been widely examined and applied to statistics and the possibility theory in recent years. Since Puri and Ralescu’s [20] introduction of the concept of fuzzy random variables, there has been growing interest in fuzzy variables [14 , 15 , 16 , 18 , 19 , 20 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30] . A number of studies [7 , 8 , 9 , 10 , 11 , 19] have investigated renewal theory in the fuzzy environment. Hwang [12] considered the stochastic process for fuzzy random variables and proved a theorem for the fuzzy rate of a fuzzy renewal process. Popova andWu [19] proposed a theorem presenting the long-run average fuzzy reward by using the strong law of large numbers. Zhao and Liu [29] and Hong [11] discussed the renewal process by considering the fuzzy inter-arrival time and proved that the expected reward per unit time is the expected value of the ratio of the reward spent in one cycle to the length of the cycle. Zhao and Tang [30] obtained some properties of a fuzzy random renewal process generated by a sequence of independent and identically distributed fuzzy random interval times based on fuzzy random theory, particularly Blackwell’s theorem for fuzzy random variables. Hong [10] considered the convergence of fuzzy random elementary renewal variables and total rewards earned by time t in the sense of the extended Hausdorff metric d and applied this method to a sequence of independent and identically distributed fuzzy random variables to prove the fuzzy random elementary renewal theorem and the fuzzy random renewal reward theorem. All of these studies used min-norm-based fuzzy operations. In general, we can consider the extension principle realized by the means of some t-norm. Such a generalized extension principle yields different operations for fuzzy numbers, in accordance with different t-norms. Recently, many studies have focused on such t-norm-based operations regarding fuzzy numbers and fuzzy random variables [3 , 4 , 5 , 6 , 7 , 22 , 23 , 24 , 25] .
Recentlt, Wang et al. [25] studied a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables, using the extension principle associated with a class of continuous Archimedean triangular norms. They discussed the fuzzy random renewal process based on the obtained limit theorems, and derived a fuzzy random elementary renewal theorem for the long-run expected renewal rate. Wang and Watada [24] also studied a renewal reward process with fuzzy random interarrival times and rewards under the T-independence associated with any continuous Archimedean t-norm. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are T-independent fuzzy variables. They derived some limit theorems in mean chance measures for fuzzy random renewal rewards and proved a fuzzy random renewal reward theorem for the long-run expected reward per unit time of the renewal reward process. In this paper, we consider the results of Wang et al. [25] and Wang and Watada [24] again. There are two main points to be improved. The first point is that when the renewal theorem and renewal rewards theorem obtained in these papers degenerate to the corresponding the classical result in stochastic renewal process, the results do not cover those of classical stochastic renewal and renewal reward theories. The second point is that some conditions relate to the membership function of the fuzzy variable and the Archimedean t-norm of the results are restrictive. This paper improves the results of Wang and Watada [24] and Wang et al. [25] from a mathematical perspective. As a consequence, our results cover the classical theory of the random elementary renewal theorem and random renewal reward theorem. We also release some conditions relating to the membership function of fuzzy variable and Archimedean t-norm of the results of Wang and Watada [24] and Wang et al. [25] and completely generalize the classical stochastic renewal theorem and renewal rewards theorem.
2. Preliminary
Let (Γ, P (Γ), Pos ) be a possibility space. As defined in [27] , a normal fuzzy variable on the real line, R is defined as a function Y : Γ → R which has a unimodal, upper semi-continuous membership function μY on the real line such that there exists a unique real number m satisfying μY ( m ) = sup xμY ( x ) = 1. The number m = m ( Y ) is called the modal value of Y .
An L-R fuzzy variable Y = ( a , α , β ) LR has a membership function from the reals into interval [0, 1] satisfying
PPT Slide
Lager Image
where L and R are non-increasing and continuous functions from [0, 1] to [0, 1] satisfying L (0) = R (0) = 1 and L (1) = R (1) = 0.
Let [ Y ] α be the α -level sets with [ Y ] α = { x R | μY α } for α ∈ (0, 1], and Y 0 = cl{ x R | μY > 0}. Let K ( R ) denote the class of nonempty compact convex subsets of R The linear structure induced by the scalar product and the Minkowski addition is that
PPT Slide
Lager Image
for all A , B K ( R ), and λ ∈ R . If dH is the Hausdorff metric on K ( R ), which for A , B K ( R ) is given by
PPT Slide
Lager Image
then ( K ( R ), dH ) is a complete and separable metric space [diamond1]. The norm of an element of K ( R ) is denoted by
PPT Slide
Lager Image
For a fuzzy variable Y and any subset D of the real numbers, the quantity
PPT Slide
Lager Image
is considered to measure the necessity of Y belonging to D (see [3] ). If D is an interval ( a , b ), we may also write Nes { a < Y < b } instead of Nes { Y | D }.
The credibility of Y belonging to D and the expected value E [ Y ] (see [18] ) are defined as
PPT Slide
Lager Image
provided that at least one of the two integrals is finite. In particular, if Y is a nonnegative fuzzy variable (i.e., Cr { Y < 0} = 0), then E [ Y ] =
PPT Slide
Lager Image
Recall that a triangular norm (t-norm for short) is a function T : [0, 1] 2 → [0, 1] such that for any x , y , z ∈ [0, 1] the following four axioms are satisfied [13] :
  • (T1) Commutativity:T(x,y) =T(y,x),
  • (T2) Associativity:T(x,T(y,z)) =T(T(x,y),z),
  • (T3) Monotonicity:T(x,y) ≤T(x,z) whenevery≤z,
  • (T4) Boundary condition:T(x, 1) =x.
The associativity ( T 2 ) allows us to extend each t-norm T in a unique way to an n-ary operation in the usual way by induction, defining for each n -tuple( x 1 , x 2 , ⋯ , xn ) ∈ [0, 1] n
PPT Slide
Lager Image
A t-norm T is said to be Archimedean if T ( x , x ) < x for all x ∈ (0, 1). It is easy to check that the minimum t-norm is not Archimedean. Moreover, from [13] , every continuous Archimedean t-norm T can be represented by a continuous and strictly decreasing function f : [0, 1] → [0,∞] with f (1) = 0 and
PPT Slide
Lager Image
for all x ∈ (0, 1), 1 ≤ i n , where f [−1] is the pseudo-inverse of f , defined by
PPT Slide
Lager Image
The function f is called the additive generator of T .
Example 2.1. Examples of continuous Archimedean t –norms with additive generators are listed as follows [13] :
( a ) Dombi t norm D , p > 0:
PPT Slide
Lager Image
with the additive generator f ( x ) = ((1 − x )/ x ) p .
( b ) Hamacher t − norm H, p ≥ 0:
PPT Slide
Lager Image
with the additive generator f ( x ) = ln(( p + (1 − p ) x )/ x ).
(c) Sklar t norm S , p > 0:
PPT Slide
Lager Image
with the additive generator f ( x ) = (1/ p )( x p − 1).
(d) Frank t norm F , p > 0, p ≠ 1:
PPT Slide
Lager Image
with the additive generator f ( x ) = log p (( p − 1)/( px − 1)).
(e) Y ager t norm Y , p > 0:
PPT Slide
Lager Image
with the additive generator f ( x ) = (1 − x ) p .
Definition 2.2 ( [2] ). Let T be a t-norm. A family of fuzzy variables { Yi , i I } is called T -independent if for any subset { i 1 , i 2 , ⋯ , in } ⊂ I with n ≥ 2,
PPT Slide
Lager Image
for any subsets B 1 , B 2 , ⋯ , Bn of R . We say two families of fuzzy variables { Yi , i I } and { Zj , j J } are mutually T -independent if for any { i 1 , i 2 , ⋯ , in } ⊂ I and { j 1 , j 2 , ⋯ , jm } ⊂ J with n , m ≥ 1, fuzzy vectors { Y i1 , ⋯ , Yin } and { Z j1 , ⋯ , Zjm } are T -independent.
For T -independent fuzzy variables Yk , 1 ≤ k m with possibility distributions μk , 1 ≤ k m , and a function g : Rm R , the possibility distribution of g ( Y 1 , Y 2 , ⋯ , Ym ) is determined via the possibility distributions μ 1 , μ 2 , ⋯ , μm as
PPT Slide
Lager Image
where T can be any general t-norm. This is the (generalized) extension principle associated with t-norm.
For example, the sum Y 1 + Y 2 +⋯ + Yn and corresponding arithmetic mean ( Y 1 + Y 2 +⋯ + Yn )/ n are the fuzzy variables as defined by
PPT Slide
Lager Image
and
PPT Slide
Lager Image
respectively.
Following Full é r and Keresztfalvi [5] [5] , if T is an arbitrary t-norm and { Yk } are normal fuzzy variables, then the equality
PPT Slide
Lager Image
holds.
Definition 2.3 ( [15] ). Suppose that (Ω,Σ, Pr) is a probability space, Fv is a collection of fuzzy variables defined on the possibility space (Γ, P (Γ), Pos ). A fuzzy random variable is a map ξ : Ω → Fv such that for any Borel subset B of R , Pos{ ξ ( ω ) ∈ B } is a measurable function of ω .
Suppose ξ is a fuzzy random variable on Ω, from the above definition, we know for each ω ∈ Ω, ξ ( ω ) is a fuzzy variable. Further, a fuzzy random variable ξ is said to be positive if for almost every ω , the fuzzy variable ξ ( ω ) is positive almost surely.
To a fuzzy random variable ξ on Ω, for each ω ∈ Ω, the expected value of the fuzzy variable ξ ( ω ), denoted by E [ ξ ( ω )], has been proved to be a random variable. Based on such fact, the expected value of the fuzzy random variable ξ is defined as the mathematical expectation of the random variable E [ ξ ( ω )].
Definition 2.4 ( [15] ). Let ξ be a fuzzy random variable given on a probability space (Ω,Σ, Pr). The expected value of ξ is defined as
PPT Slide
Lager Image
Definition 2.5 ( [16] ). Let ξ be a fuzzy random variable, and B be a Borel subset of R . The mean chance of an event ξ B is defined as
PPT Slide
Lager Image
The expected value (2) is equivalent to the following form :
PPT Slide
Lager Image
A sequence of fuzzy random variables { ξn } is said to be uniformly and essentially bounded if there are two real numbers bL and bU such that for each k = 1, 2, ⋯ , we have Ch { bL ξn bU } = 1. Moreover, we have the following convergence mode for a sequence of fuzzy random variables.
3. Fuzzy random renewal and renewal rewards process
Let ξn , n = 1, 2,⋯ be a sequence of fuzzy random variables defined on a probability space (Ω,Σ, Pr). For each n , we denote ξn as the interarrival time between the ( n − 1)th and the nth event. Define
PPT Slide
Lager Image
It is clear that Sn is the time when the nth renewal occurs, and for any given ω ∈ Ω and integer n , Sn ( ω ) = ξ 1 ( ω ) +⋯ + ξn ( ω ) is a fuzzy variable. Let N ( t ) denote the total number of the events that have occurred by time t . Then we have
PPT Slide
Lager Image
For any ω ∈ Ω, N ( t )( ω ) = max{ n | Sn ( ω ) ≤ t } is a nonnegative integer-valued fuzzy variable on the possibility space (Γ, P (Γ), Pos ). That is, N ( t )( ω )( γ ) is a nonnegative integer for any γ ∈ Γ, ω ∈ Ω and t > 0. We call N ( t ) a fuzzy random renewal variable, and the process { N ( t ), t > 0} a fuzzy random renewal process.
On the basis of the above renewal process { N ( t ), t > 0} with fuzzy random interarrival times ξn , n ≥ 1, suppose each time a renewal occurs we receive a reward which is a fuzzy random variable. We denote ηn as the reward earned at each time of the nth renewal. Let C ( t ) represent the total reward earned by time t , then we have
PPT Slide
Lager Image
where N ( t ) is the fuzzy random renewal variable.
In the paper of Wang and Watada [24] , and Wang et al. [25] , they discussed fuzzy stochastic renewal theories within the following conditions:
A1. T can be any continuous Archimedean t-norm with additive generator f . The interarrival times { ξn } and rewards { ηn } are positive fuzzy random variables, and for almost every ω ∈ Ω, { ξn ( ω )} and { ηn ( ω )} are T -independent fuzzy variable sequences, respectively.
A2. Π is a nonnegative unimodal real function with Π(0) = 1 and Π(− r ) = 0, where r can be any positive real number. The support of Π is denoted by
PPT Slide
Lager Image
.
A3. The convex hull of composition function f ◦ Π in the support
PPT Slide
Lager Image
of Π satisfies:
PPT Slide
Lager Image
for any nonzero r
PPT Slide
Lager Image
.
Their two main results are the following.
Theorem 3.1 (Fuzzy random elementary renewal theorem [25] ). Suppose ξ 1 , ξ 2 ,⋯ is a sequence of fuzzy random inter-arrival times where μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) for almost every ω ∈ Ω k = 1, 2,⋯ with Π I (− a ) = 0, a > 0, and N ( t ) is the fuzzy random renewal variable. If { Uk } is i.i.d. random variable sequences with a finite expected value and Uk a + h almost surely, then we have
PPT Slide
Lager Image
Theorem 3.2 (Fuzzy random renewal reward theorem [24] ). Suppose ( ξ 1 , η 1 ), ( ξ 2 , η 2 ),⋯ is a sequence of pairs of fuzzy random inter-arrival times and rewards, where { ηk } is uniformly essentially bounded, μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) with Uk a + h almost surely, and μ ηk(ω) ( x ) = Π R ( x Vk ( ω )) , for almost every ω ∈ Ω, k = 1, 2, ⋯ , N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { Uk } and { Vk } are i.i.d. random variable sequences with finite expected values, respectively, and { ξk ( ω )} and { ηk ( ω )} are mutually Tindependent for almost every ω ∈ Ω , then
PPT Slide
Lager Image
There are several remarks regarding these results which can be improved.
Remark 3.1. In the assumption A1. the interarrival times { ξn } and rewards { ηn } need not be the same T -independent fuzzy variable sequences, respectively.
Remark 3.2. The condition Uk a + h in Theorem 3.1 is strong. If { ξk , k ≥ 1} degenerates to a sequence of i.i.d. random variables, then the result in Theorem 1 does not cover the elementary renewal theorem in the stochastic case (see [21] )
Remark 3.3. The condition Uk a + h in Theorem 3.2 is strong. If {( ξk , ηk ), k ≥ 1} degenerates to a sequence of i.i.d. random variables, then the result in Theorem 3.2 does not cover the renewal reward theorem in the stochastic case (see [21] )
We consider the condition A3.
PPT Slide
Lager Image
for any nonzero r
PPT Slide
Lager Image
.
Example 3.3. Let Π be the possibility distribution of a normal fuzzy variable with Π( x ) = e x2 , x R , Yp ( x , y ) is a Yager t −norm with the additive generator f ( x ) = (1− x ) p . Then f ◦Π( x ) = (1− e x2 ) p on
PPT Slide
Lager Image
= R and hence
PPT Slide
Lager Image
= 0, x
PPT Slide
Lager Image
.
Example 3.4. Let Π be the possibility distribution of a normal fuzzy variable with Π( x ) = e −|x|λ , 0 < λ < 1, x R , H 1 ( x , y ) is a Hamacher t −norm with the additive generator f ( x ) = − lnx . Then f ◦ Π( x ) = | x | λ , 0 < λ < 1 on
PPT Slide
Lager Image
= R and hence co( f ◦ Π)
PPT Slide
Lager Image
( x ) = 0, x
PPT Slide
Lager Image
.
Remark 3.4. As we can see in the Examples 3.3 and 3.4, the condition A3. is strong. This assumption can be released.
Remark 3.5. The condition that { ηk } is uniformly essentially bounded in Theorem 2 can be released.
Throughout this paper, we discuss fuzzy stochastic renewal and renewal rewards theories within the following conditions:
H1. Ti , i = 1, 2 can be any continuous Archimedean t-norms with additive generators fi , i = 1, 2. The interarrival times { ξn } and rewards { ηn } are positive fuzzy random variables, and for almost every ω ∈ Ω, { ξn ( ω )} is T 1 -independent fuzzy variable sequences and { ηn ( ω )} is T 2 -independent fuzzy variable sequences, respectively.
H2. Π is a nonnegative unimodal real function with Π(0) = 1, Π( x ) < 1, x ≠ 0,
PPT Slide
Lager Image
3.1. Fuzzy random elementary renewal theorem . We first consider the following result.
Proposition 3.5. et T be a continuous Archimedean t-norms with additive generators f. Let { Yk } be a sequence of T-independent fuzzy variables with membership function μk ( x ) = Π( x uk ) and let Sn = Y 1 +· · ·+ Yn , m ( Sn ) = u 1 +· · ·+ un .
Then, for all ϵ > 0 we have
PPT Slide
Lager Image
Proof . Define
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
is an upper semi-continuous nonnegative unimodal real function with
PPT Slide
Lager Image
, r ≠ 0 and Π( x ) ≤
PPT Slide
Lager Image
, x R . Let
PPT Slide
Lager Image
be fuzzy variable with membership function
PPT Slide
Lager Image
and let
PPT Slide
Lager Image
. Since
PPT Slide
Lager Image
and, for all ϵ > 0
PPT Slide
Lager Image
it suffices to prove that
PPT Slide
Lager Image
And to show this fact, it is also easy to check that, for any 0 < α < 1
PPT Slide
Lager Image
Let 0 < α < 1, δ > 0 be given and f (1− δ ) = t 0 . Suppose that
PPT Slide
Lager Image
and let Hα,δ = { k | f ( αk ) > t 0 }. Then, the number of Hα,δ is less than or equal to N where N is the smallest natural number bigger than f ( α )/ t 0 . We note that if k Hα,δ , then αk ≥ 1 − δ and if k Hα,δ , then αk α Then, we have from (1)
PPT Slide
Lager Image
where B (0, tα ) = { x : | x | ≤ tα }, tα =
PPT Slide
Lager Image
= max{− lα , rα } such that
PPT Slide
Lager Image
and the second equality above comes from the convexity of [ A ] α . Hence, we have, for large n ,
PPT Slide
Lager Image
The first term goes to 0 because δ is arbitrary and
PPT Slide
Lager Image
by the assumption H2 and the second term goes to 0 as n → ∞. This completes the proof. □
Following the line of the proof in regarding to the Theorem of Hong [8] and using Proposition 3.5, the following result is obtained.
Proposition 3.6. Let T be a continuous Archimedean t-norms with additive generators f. Let { Yk } be a sequence of a T-independent fuzzy variable with the membership function μk ( x ) = Π( x uk ) and let Sn = Y 1 +⋯ + Yn , m ( Sn ) = u 1 +· · ·+ un. If μk a , k = 1, 2,⋯ and ( u 1 +· · ·+ un )/ n u as n → ∞ , then we have, for all ϵ > 0
PPT Slide
Lager Image
Note 1. It is noted that since
PPT Slide
Lager Image
the condition
PPT Slide
Lager Image
implies
PPT Slide
Lager Image
From Note 1, we have the following result:
Proposition 3.7. Let T be a continuous Archimedean t-norms with additive generators f. Let { Yk } be a sequence of a T-independent fuzzy variable with the membership function μk ( x ) = Π( x uk ) and let Sn = Y 1 +⋯ + Yn , m ( Sn ) = u 1 +· · ·+ un. If μk a , k = 1, 2,⋯ and ( u 1 +· · ·+ un )/ n u as n → ∞, then we have, for all ϵ > 0
PPT Slide
Lager Image
The following is a well-known result of probability theory [1] .
Lemma 3.8 ( [1] ). Let
PPT Slide
Lager Image
be independent and identically distributed (i.i.d.) random variables. Then E | X 1 | < ∞ iff n −1 max 1≤kn | Xk | → 0 a.s. iff
PPT Slide
Lager Image
a.s .
The following result is immediate from Proposition 3.7 and Lemma 3.8.
Proposition 3.9. Suppose ξ 1 , ξ 2 ,⋯ is a sequence of fuzzy random inter-arrival times where μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) for almost every ω ∈ Ω k = 1, 2,⋯ with Π I (− a ) = 0, a > 0, and N ( t ) is the fuzzy random renewal variable. If { Uk } is i.i.d. random variable sequences with finite expected value and Uk a almost surely, then we have
PPT Slide
Lager Image
Lemma 3.10 ( [25] ). Suppose ξ 1 , ξ 2 ,⋯ is a sequence of fuzzy random interarrival times where μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) for almost every ω ∈ Ω k = 1, 2,⋯ with Π I (− a ) = 0, a > 0, and N ( t ) is the fuzzy random renewal variable. If Uk a almost surely, then we have
PPT Slide
Lager Image
where M is the smallest integer such that M rt and ξ is a fuzzy variable with μξ ( x ) = Π I ( x ).
Definition 3.11 ( [16] ). A sequence { ξn } of fuzzy variables is said to converge in credibility to a fuzzy variable ξ , if for every ε > 0,
PPT Slide
Lager Image
Theorem 3.12. Suppose ξ 1 , ξ 2 ,⋯ is a sequence of fuzzy random inter-arrival times where μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) for almost every ω ∈ Ω k = 1, 2,⋯ with Π I (− a ) = 0, a > 0, and N ( t ) is the fuzzy random renewal variable. If { Uk } is a sequences of i.i.d. random variables with finite expected value and Uk a almost surely, then we have
PPT Slide
Lager Image
Proof . According to Proposition 3.9, we know that { N ( t )( ω )/ t } converges in credibility to 1/ E [ U 1 ] almost surely. Then, it follows from Liu [17 , Theorem 3.59] that for any r R ,
PPT Slide
Lager Image
Let E [ U 1 ] − a = ϵ > 0. Since
PPT Slide
Lager Image
almost surely, we note that, for almost all ( ω ), there exists T ( ω ) > 0, depending on ω such that for M T ( ω ),
PPT Slide
Lager Image
. By Lemma 3.10, for r > 2/ ϵ
PPT Slide
Lager Image
As a consequence, by the Lebesgue dominated convergence theorem, we have
PPT Slide
Lager Image
Here, we define an order of two fuzzy variables. Let A , B be two fuzzy variables. We define A B iff Cr { A r } ≤ Cr { B r } for all real number r .
Lemma 3.13. Suppose ξ 1 , ξ 2 ,⋯ is a sequence of fuzzy random inter-arrival times where μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) for almost every ω ∈ Ω k = 1, 2,⋯ with Π I (− a ) = 0, a > 0, and N ( t ) is the fuzzy random renewal variable. If { Uk } is a sequences of i.i.d. random variables with finite expected value and Uk a almost surely, then we have
PPT Slide
Lager Image
Proof . Since P { Uk = a } < 1, there exists δ > 0 such that P { Uk a + δ } = p > 0 for all n = 1, 2, 3,⋯ . Define
PPT Slide
Lager Image
and
PPT Slide
Lager Image
and N′ ( t ) be the corresponding quantities for the sequence
PPT Slide
Lager Image
. It is obvious that, considering
PPT Slide
Lager Image
as degenerated fuzzy random variables such that
PPT Slide
Lager Image
and N′ ( t ) ≥ N ( t ) for each t , and hence we have that
PPT Slide
Lager Image
Since the random variables
PPT Slide
Lager Image
are independent with a Bernoullian distribution, elementary computations show that
PPT Slide
Lager Image
Hence we have, δ being fixed,
PPT Slide
Lager Image
Theorem 3.14 ( [1] ). Let { Xn } is a sequence of random variables such that supnE | Xn | p = M < ∞ for some p > 0. If Xn X in distribution, then for each r < p :
PPT Slide
Lager Image
Our first main result is the following which improves Theorem 3.1.
Theorem 3.15 (Fuzzy random elementary renewal theorem). Suppose ξ 1 , ξ 2 ,⋯ is a sequence of fuzzy random inter-arrival times where μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) k = 1, 2,⋯ for almost every ω ∈ Ω, with Π I (− a ) = 0, a > 0, and N ( t ) is the fuzzy random renewal variable. If { Uk } is a sequence of i.i.d. random variables with finite expected value and Uk a almost surely, then we have
PPT Slide
Lager Image
Proof . By Lemma 3.13, we have
PPT Slide
Lager Image
Since Theorem 3.12 implies
PPT Slide
Lager Image
an application of Theorem 3.14 with Xn ( ω ) = E [ N (n)( ω )]/ n and p = 2 yield (2) with t replaced by n in (2), from which (2) itself follows at once. □
Remark 3.6. If { ξk , k ≥ 1} degenerates to a sequences of i.i.d. random variables, the result in Theorem 3.15 is just the renewal theorem in the stochastic case (see [21] ).
3.2. Fuzzy random renewal reward theorem. A slight modification of the result regarding the Theorem of Hong [8] using Lemma 3.8 and Proposition 3.5, gives the following result.
Proposition 3.16. Suppose ( ξ 1 , η 1 ), ( ξ 2 , η 2 ),⋯ is a sequence of pairs of fuzzy random interarrival times and rewards, where μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) with Π I (− a ) = 0, Uk ( ω ) ≥ a > 0 almost surely, and μ ηk(ω) ( x ) = Π R ( x Vk ( ω )), k = 1, 2, ⋯ , N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { Uk } and { Vk } are i.i.d. random variable sequences with finite expected values, respectively, then for any ϵ > 0,
PPT Slide
Lager Image
Lemma 3.17. Suppose ( ξ 1 , η 1 ), ( ξ 2 , η 2 ),⋯ is a sequence of pairs of fuzzy random interarrival times and rewards, where μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) with Π I (− a ) = 0, Uk ( ω ) ≥ a > 0 almost surely, and μ ηk(ω) ( x ) = Π R ( x Vk ( ω )), k = 1, 2, ⋯ , , with
PPT Slide
Lager Image
, N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { Uk } and { Vk } are i.i.d. random variable sequences with E [ U 1 ] < ∞ and E [| V 1 |] < ∞ , then there exists a function
PPT Slide
Lager Image
, r ≥ 0 and a function
PPT Slide
Lager Image
, r ≤ 0 almost surely such that for big t > 0,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Proof . Let N′ ( t ) be the renewal random variables defined in Lemma 3. We consider that for 0 < l 1 < 1/ =
PPT Slide
Lager Image
< l 2 , and r ≥ 0,
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Then, by the classical renewal theorem, lim t→∞ N′ ( t )( ω )/ t =
PPT Slide
Lager Image
almost surely, we have for big t > 0,
PPT Slide
Lager Image
Nest, we consider that
PPT Slide
Lager Image
where η is a fuzzy variable with μη ( x ) = Π R ( x ). Then, we have for big t > 0, for some ϵ > 0,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
By similar argument, we have for big t > 0, for some ϵ > 0,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Theorem 3.18. Suppose ( ξ 1 , η 1 ), ( ξ 2 , η 2 ),⋯ is a sequence of pairs of fuzzy random interarrival times and rewards, where { ηk } is μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) with Π I (− a ) = 0, Uk ( ω ) ≥ a > 0 almost surely, and μ ηk(ω) ( x ) = Π R ( x Vk ( ω )), for almost every ω ∈ Ω with
PPT Slide
Lager Image
, k = 1, 2, ⋯ , N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { Uk } and { Vk } are i.i.d. random variable sequences with E [ U 1 ] < ∞ and E [| V 1 |] < ∞ , then
PPT Slide
Lager Image
Proof . According to Proposition 3.16, we know that { C ( t )( ω )/ t } converges in credibility to E [ V 1 ]/ E [ U 1 ] almost surely. It follows from Liu [17 , Theorem3.59] that for almost all r ≥ 0,
PPT Slide
Lager Image
and for almost all r ≤ 0,
PPT Slide
Lager Image
As a consequence, by Lemma 3.17 and the Lebesgue dominated convergence theorem, we have
PPT Slide
Lager Image
Theorem 3.19 ( [1] ). Let { Xn } be a sequence of random variables such that E | Xn | < ∞, n = 1, 2, 3,⋯ and Xn X in probability. Then the following two propositions are equivalent:
i) {| Xn |} is uniformly integrable
ii)E | Xn | → E | X |.
Our second main result is the following which improves Theorem 3.2.
Theorem 3.20 (Fuzzy random renewal reward theorem). Suppose ( ξ 1 , η 1 ), ( ξ 2 , η 2 ),⋯ is a sequence of pairs of fuzzy random interarrival times and rewards, where { ηk } is μ ξk(ω) ( x ) = Π I ( x Uk ( ω )) with Π I (− a ) = 0, Uk ( ω ) ≥ a > 0 almost surely, and μ ηk(ω) ( x ) = Π R ( x Vk ( ω )) , for almost every ω ∈ Ω with
PPT Slide
Lager Image
, k = 1, 2, ⋯ , N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { Uk } and { Vk } are i.i.d. random variable sequences with E [ U 1 ] < ∞ and E [| V 1 |] < ∞ , then
PPT Slide
Lager Image
Proof . Let N′ ( t ) is the random renewal variable defined in Lemma 3.13. Let Π R′ be a function with Π R′ ( x ) = Π R ( x ) if x ≥ 0 and 0, otherwise, and let
PPT Slide
Lager Image
be fuzzy random variables with μ η′k(ω) ( x ) = Π R′ ( x Vk ( ω )), for almost every ω ∈ Ω. Similarly, we define Π R′′ to be a function with Π R′′ ( x ) = Π R ( x ) if x ≤ 0 and 0, otherwise, and
PPT Slide
Lager Image
to be fuzzy random variables with μ η′′k(ω) ( x ) = Π R′′ ( x Vk ( ω )), for almost every ω ∈ Ω. Then we clearly have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
It is also noted that
PPT Slide
Lager Image
for some positive real numbers v′ , v′′ > 0, respectively. Then we have
PPT Slide
Lager Image
Similarly, we have
PPT Slide
Lager Image
Then, since
PPT Slide
Lager Image
, k = 1, 2,⋯ is a sequence of pairs of i.i.d. random variables, by the classical stochastic renewal reward theorem, we have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Then, by Theorem 3.19, the class of random variables
PPT Slide
Lager Image
is uniformly integrable, and hence
PPT Slide
Lager Image
are uniformly integrable. Since
PPT Slide
Lager Image
{ E [ C ( t )( ω )]/ t } is also uniformly integrable. Since, by Theorem 3.18, we have
PPT Slide
Lager Image
and hence by Theorem 3.19, again we have
PPT Slide
Lager Image
which completes the proof. □
Remark 3.7. If {( ξk , ηk ), k ≥ 1} degenerates to a sequences of pair of i.i.d. random variables, the result in Theorem 3.20 is simply the renewal rewards theorem in stochastic case (see [21] ).
4. Conclusion
In this paper, we considered the random fuzzy elementary renewal theorem and the random fuzzy renewal reward theorem studied by Wang et al. [25] and Wang and Watada [24] when fuzzy random variables are T -independent and improved the results from a mathematical perspective. If the sequence of fuzzy random variables degenerates to a sequence of random variables, the main results are just the renewal theorem and the renewal rewards theorem in the stochastic case (see [21] ).
BIO
Jae Duck Kim received the B.S., M.S. and Ph. D degrees in mathematics from Myongji University in 2006, 2008 and 2010, respectively. Since 2015, he has been a Professor in BangMok College of Basic Studies, Myongji University, Korea.
Department of Mathematics, Myongji University, Yongin 449-728, South Korea.
e-mail: jdkim@mju.ac.kr
Dug Hun Hong received the B.S., M.S. degrees in mathematics from Kyungpook National University, Taegu, Korea and Ph. D degree in mathematics from University of Minesota, Twin City in 1981, 1983 and 1990, respectively. From 1991 to 2003, he worked with de-partment of Statistics and School of Mechanical and Automotive Engineering, Catholic University of Daegu, Daegu, Korea. since 2003, he has been a Professor in Department of Mathematics, Myongji University, Korea. His research interests include general fuzzy theory with application and probability theory.
Department of Mathematics, Myongji University, Yongin 449-728, South Korea.
e-mail: dhhong@mju.ac.kr
References
Chung K.L. 1974 A course in probability theory second edition Academic Press New York and London
De Cooman G. (1997) Possibility theory III International Journal of General Systems 25 352 - 371
Fullér R. (1992) A law of large numbers for fuzzy numbers Fuzzy Sets and Systems 45 299 - 303    DOI : 10.1016/0165-0114(92)90147-V
Fullér R. , Triesch E. (1993) A note on the law of large numbers for fuzzy variables Fuzzy Sets and Systems 55 235 - 236    DOI : 10.1016/0165-0114(93)90136-6
Fullér R. , Keresztfalvi T. (1991) On generalization of Nguyen's theorem Fuzzy Sets and Systems 41 371 - 374    DOI : 10.1016/0165-0114(91)90139-H
Hong D.H. , Lee J. (2001) On the law of large numbers for mutually T-related fuzzy numbers Fuzzy Sets and Systems 121 537 - 543    DOI : 10.1016/S0165-0114(99)00136-0
Hong D.H. , Ahn C.H. (2003) Equivalent conditions for laws of large numbers for T-related L-R fuzzy numbers Fuzzy Sets and Systems 136 387 - 395    DOI : 10.1016/S0165-0114(02)00217-8
Hong D.H. (2006) Renewal process with T-related fuzzy inter-arrival times and fuzzy rewards Information Sciences 176 2386 - 2395    DOI : 10.1016/j.ins.2005.06.008
Hong D.H. (2010) Blackwell's Theorem for T-related fuzzy variables Information Sciences 180 1769 - 1777    DOI : 10.1016/j.ins.2010.01.006
Hong D.H. (2010) Uniform convergence of fuzzy random renewal process Fuzzy Optimization and Decision Making 9 275 - 288    DOI : 10.1007/s10700-010-9085-y
Hong D.H. (2007) Renewal process for fuzzy variables International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15 493 - 501    DOI : 10.1142/S0218488507004819
Hwang C-M. (2000) A theorem of renewal process for fuzzy random variables and its application Fuzzy Sets and Systems 116 237 - 244    DOI : 10.1016/S0165-0114(98)00143-2
Klement E.P. , Mesiar R. , Pap E. 2000 Triangular norms, Trends in Logic, Vol. 8 Kluwer Dordrecht
Kwakernaak H. (1978) Fuzzy random variables I: Definitions and theorems Information Sciences 15 1 - 29    DOI : 10.1016/0020-0255(78)90019-1
Liu Y.-K. , Liu B. (2003) Fuzzy random variable: A scalar expected value operator Fuzzy Opti-mization and Decision Making 2 143 - 160    DOI : 10.1023/A:1023447217758
Liu Y.-K. , Liu B. (2005) On minimum-risk problems in fuzzy random decision systems Com-puters and Operations Research 32 257 - 283    DOI : 10.1016/S0305-0548(03)00235-1
Liu B. 2004 Uncertain Theory: An Introduction to its Axiomatic Foundations Springer-Verlag Berlin
Liu B. , Liu Y.-K. (2002) Expected value of a fuzzy variable and fuzzy expected value models IEEE Transaction on Fuzzy Systems 10 445 - 450    DOI : 10.1109/TFUZZ.2002.800692
Popova E. , Wu H. C. (1999) Renewal reward processes with fuzzy rewards and their applica-tions to T-age replacement policies European Journal of Operational Research 117 606 - 617    DOI : 10.1016/S0377-2217(98)00247-1
Puri M.L. , Ralescu D.A. (1986) Fuzzy random variables Journal of Mathematical Analysis and Applications 114 409 - 422    DOI : 10.1016/0022-247X(86)90093-4
Ross S.M. 1996 Stochastic Processes John Wiley and Sons New York
Terán P. (2008) Strong law of large numbers for t-normed arithmetics Fuzzy Sets and Systems 159 343 - 360    DOI : 10.1016/j.fss.2007.06.006
Triesch E. (1993) Characterization of Archimedean t-norms and a law of large numbers Fuzzy Sets and Systems 58 339 - 342    DOI : 10.1016/0165-0114(93)90507-E
Wang S. , Watada J. (2009) Fuzzy random renewal reward process and its applications Information Sciences 179 4057 - 4069    DOI : 10.1016/j.ins.2009.08.016
Wang S. , Liu Y-K. , Watada J. (2009) Fuzzy random renewal process with queueing applications Computers and Mathematics with Applications 57 1232 - 1248    DOI : 10.1016/j.camwa.2009.01.030
Wang S. , Watada J. (2009) Studying distribution functions of fuzzy random variables and its applications to critical value functions International Journal of Innovative Computing, Information and Control 5 2 279 - 292
Zadeh L.A. (1978) Fuzzy sets as a basis for a theory of possibility Fuzzy Sets and Systems 1 3 - 28    DOI : 10.1016/0165-0114(78)90029-5
Zadeh L.A. (1965) Fuzzy sets Information and Control 8 338 - 353    DOI : 10.1016/S0019-9958(65)90241-X
Zhao R. , Liu B. (2003) Renewal process with fuzzy inter-arrival times and rewards Interna-tional Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11 573 - 586    DOI : 10.1142/S0218488503002338
Zhao R. , Tang W. (2006) Some properties of fuzzy random processes IEEE Transactions on Fuzzy Systems 2 173 - 179    DOI : 10.1109/TFUZZ.2005.864088
Zhao R. , Tang W. , Wang C. (2007) Fuzzy random renewal process and renewal reward process Fuzzy Optimization and Decision Making 6 279 - 295    DOI : 10.1007/s10700-007-9012-z