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. 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. 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 μ_Y on the real line such that there exists a unique real number m satisfying μ_Y ( m ) = sup _xμ_Y ( 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 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 | μ_Y ≥ α } for α ∈ (0, 1], and Y ₀ = cl{ x ∈ R | μ_Y > 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 for all A , B ∈ K ( R ), and λ ∈ R . If d_H is the Hausdorff metric on K ( R ), which for A , B ∈ K ( R ) is given by then ( K ( R ), d_H ) is a complete and separable metric space [diamond1]. The norm of an element of K ( R ) is denoted by For a fuzzy variable Y and any subset D of the real numbers, the quantity 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 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 ] = 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] : 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 , ⋯ , x_n ) ∈ [0, 1] ⁿ 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 for all x ∈ (0, 1), 1 ≤ i ≤ n , where f ^[−1] is the pseudo-inverse of f , defined by 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: with the additive generator f ( x ) = ((1 − x )/ x ) ^p . ( b ) Hamacher t − norm H, p ≥ 0: with the additive generator f ( x ) = ln(( p + (1 − p ) x )/ x ). (c) Sklar t − norm S , p > 0: with the additive generator f ( x ) = (1/ p )( x ^−p − 1). (d) Frank t − norm F , p > 0, p ≠ 1: with the additive generator f ( x ) = log _p (( p − 1)/( p^x − 1)). (e) Y ager t − norm Y , p > 0: with the additive generator f ( x ) = (1 − x ) ^p . Definition 2.2 ( [2] ). Let T be a t-norm. A family of fuzzy variables { Y_i , i ∈ I } is called T -independent if for any subset { i ₁ , i ₂ , ⋯ , i_n } ⊂ I with n ≥ 2, for any subsets B ₁ , B ₂ , ⋯ , B_n of R . We say two families of fuzzy variables { Y_i , i ∈ I } and { Z_j , j ∈ J } are mutually T -independent if for any { i ₁ , i ₂ , ⋯ , i_n } ⊂ I and { j ₁ , j ₂ , ⋯ , j_m } ⊂ J with n , m ≥ 1, fuzzy vectors { Y _i1 , ⋯ , Y_in } and { Z _j1 , ⋯ , Z_jm } are T -independent. For T -independent fuzzy variables Y_k , 1 ≤ k ≤ m with possibility distributions μ_k , 1 ≤ k ≤ m , and a function g : R^m → R , the possibility distribution of g ( Y ₁ , Y ₂ , ⋯ , Y_m ) is determined via the possibility distributions μ ₁ , μ ₂ , ⋯ , μ_m as 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 +⋯ + Y_n and corresponding arithmetic mean ( Y ₁ + Y ₂ +⋯ + Y_n )/ n are the fuzzy variables as defined by and respectively. Following Full é r and Keresztfalvi [5] [5] , if T is an arbitrary t-norm and { Y_k } are normal fuzzy variables, then the equality holds. Definition 2.3 ( [15] ). Suppose that (Ω,Σ, Pr) is a probability space, F_v is a collection of fuzzy variables defined on the possibility space (Γ, P (Γ), Pos ). A fuzzy random variable is a map ξ : Ω → F_v 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 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 The expected value (2) is equivalent to the following form : A sequence of fuzzy random variables { ξ_n } is said to be uniformly and essentially bounded if there are two real numbers b_L and b_U such that for each k = 1, 2, ⋯ , we have Ch { b_L ≤ ξ_n ≤ b_U } = 1. Moreover, we have the following convergence mode for a sequence of fuzzy random variables. Let ξ_n , n = 1, 2,⋯ be a sequence of fuzzy random variables defined on a probability space (Ω,Σ, Pr). For each n , we denote ξ_n as the interarrival time between the ( n − 1)th and the nth event. Define It is clear that S_n is the time when the nth renewal occurs, and for any given ω ∈ Ω and integer n , S_n ( ω ) = ξ ₁ ( ω ) +⋯ + ξ_n ( ω ) is a fuzzy variable. Let N ( t ) denote the total number of the events that have occurred by time t . Then we have For any ω ∈ Ω, N ( t )( ω ) = max{ n | S_n ( 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 ξ_n , n ≥ 1, suppose each time a renewal occurs we receive a reward which is a fuzzy random variable. We denote η_n 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 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 { ξ_n } and rewards { η_n } are positive fuzzy random variables, and for almost every ω ∈ Ω, { ξ_n ( ω )} and { η_n ( 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 . A3. The convex hull of composition function f ◦ Π in the support of Π satisfies: for any nonzero r ∈ . 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 − U_k ( 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 { U_k } is i.i.d. random variable sequences with a finite expected value and U_k ≥ a + h almost surely, then we have 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 { η_k } is uniformly essentially bounded, μ _ξk(ω) ( x ) = Π _I ( x − U_k ( ω )) with U_k ≥ a + h almost surely, and μ _ηk(ω) ( x ) = Π _R ( x − V_k ( 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 { U_k } and { V_k } are i.i.d. random variable sequences with finite expected values, respectively, and { ξ_k ( ω )} and { η_k ( ω )} are mutually Tindependent for almost every ω ∈ Ω , then There are several remarks regarding these results which can be improved. Remark 3.1. In the assumption A1. the interarrival times { ξ_n } and rewards { η_n } need not be the same T -independent fuzzy variable sequences, respectively. Remark 3.2. The condition U_k ≥ a + h in Theorem 3.1 is strong. If { ξ_k , 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 U_k ≥ a + h in Theorem 3.2 is strong. If {( ξ_k , η_k ), 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. for any nonzero r ∈ . Example 3.3. Let Π be the possibility distribution of a normal fuzzy variable with Π( x ) = e ^−x2 , x ∈ R , Y_p ( 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 = R and hence = 0, x ∈ . 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 = R and hence co( f ◦ Π) ( x ) = 0, x ∈ . 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 { η_k } 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. T_i , i = 1, 2 can be any continuous Archimedean t-norms with additive generators f_i , i = 1, 2. The interarrival times { ξ_n } and rewards { η_n } are positive fuzzy random variables, and for almost every ω ∈ Ω, { ξ_n ( ω )} is T ₁ -independent fuzzy variable sequences and { η_n ( 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, 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 { Y_k } be a sequence of T-independent fuzzy variables with membership function μ_k ( x ) = Π( x − u_k ) and let S_n = Y ₁ +· · ·+ Y_n , m ( S_n ) = u ₁ +· · ·+ u_n . Then, for all ϵ > 0 we have Proof . Define Then is an upper semi-continuous nonnegative unimodal real function with , r ≠ 0 and Π( x ) ≤ , x ∈ R . Let be fuzzy variable with membership function and let . Since and, for all ϵ > 0 it suffices to prove that And to show this fact, it is also easy to check that, for any 0 < α < 1 Let 0 < α < 1, δ > 0 be given and f (1− δ ) = t ₀ . Suppose that and let H_α,δ = { k | f ( α_k ) > t ₀ }. Then, the number of H_α,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 ∉ H_α,δ , then α_k ≥ 1 − δ and if k ∈ H_α,δ , then α_k ≥ α Then, we have from (1) where B (0, t_α ) = { x : | x | ≤ t_α }, t_α = = max{− l_α , r_α } such that and the second equality above comes from the convexity of [ A ] _α . Hence, we have, for large n , The first term goes to 0 because δ is arbitrary and 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 { Y_k } be a sequence of a T-independent fuzzy variable with the membership function μ_k ( x ) = Π( x − u_k ) and let S_n = Y ₁ +⋯ + Y_n , m ( S_n ) = u ₁ +· · ·+ u_n. If μ_k ≥ a , k = 1, 2,⋯ and ( u ₁ +· · ·+ u_n )/ n → u as n → ∞ , then we have, for all ϵ > 0 Note 1. It is noted that since the condition implies From Note 1, we have the following result: Proposition 3.7. Let T be a continuous Archimedean t-norms with additive generators f. Let { Y_k } be a sequence of a T-independent fuzzy variable with the membership function μ_k ( x ) = Π( x − u_k ) and let S_n = Y ₁ +⋯ + Y_n , m ( S_n ) = u ₁ +· · ·+ u_n. If μ_k ≥ a , k = 1, 2,⋯ and ( u ₁ +· · ·+ u_n )/ n → u as n → ∞, then we have, for all ϵ > 0 The following is a well-known result of probability theory [1] . Lemma 3.8 ( [1] ). Let be independent and identically distributed (i.i.d.) random variables. Then E | X ₁ | < ∞ iff n ⁻¹ max _1≤k≤n | X_k | → 0 a.s. iff 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 − U_k ( 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 { U_k } is i.i.d. random variable sequences with finite expected value and U_k ≥ a almost surely, then we have 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 − U_k ( 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 U_k ≥ a almost surely, then we have where M is the smallest integer such that M ≥ rt and ξ is a fuzzy variable with μ_ξ ( x ) = Π _I ( x ). Definition 3.11 ( [16] ). A sequence { ξ_n } of fuzzy variables is said to converge in credibility to a fuzzy variable ξ , if for every ε > 0, 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 − U_k ( 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 { U_k } is a sequences of i.i.d. random variables with finite expected value and U_k ≥ a almost surely, then we have 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 , Let E [ U ₁ ] − a = ϵ > 0. Since almost surely, we note that, for almost all ( ω ), there exists T ( ω ) > 0, depending on ω such that for M ≥ T ( ω ), . By Lemma 3.10, for r > 2/ ϵ As a consequence, by the Lebesgue dominated convergence theorem, we have 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 − U_k ( 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 { U_k } is a sequences of i.i.d. random variables with finite expected value and U_k ≥ a almost surely, then we have Proof . Since P { U_k = a } < 1, there exists δ > 0 such that P { U_k ≥ a + δ } = p > 0 for all n = 1, 2, 3,⋯ . Define and and N′ ( t ) be the corresponding quantities for the sequence . It is obvious that, considering as degenerated fuzzy random variables such that and N′ ( t ) ≥ N ( t ) for each t , and hence we have that Since the random variables are independent with a Bernoullian distribution, elementary computations show that Hence we have, δ being fixed, Theorem 3.14 ( [1] ). Let { X_n } is a sequence of random variables such that sup_nE | X_n | ^p = M < ∞ for some p > 0. If X_n → X in distribution, then for each r < p : 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 − U_k ( 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 { U_k } is a sequence of i.i.d. random variables with finite expected value and U_k ≥ a almost surely, then we have Proof . By Lemma 3.13, we have Since Theorem 3.12 implies an application of Theorem 3.14 with X_n ( 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 { ξ_k , 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 − U_k ( ω )) with Π _I (− a ) = 0, U_k ( ω ) ≥ a > 0 almost surely, and μ _ηk(ω) ( x ) = Π _R ( x − V_k ( ω )), k = 1, 2, ⋯ , N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { U_k } and { V_k } are i.i.d. random variable sequences with finite expected values, respectively, then for any ϵ > 0, 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 − U_k ( ω )) with Π _I (− a ) = 0, U_k ( ω ) ≥ a > 0 almost surely, and μ _ηk(ω) ( x ) = Π _R ( x − V_k ( ω )), k = 1, 2, ⋯ , , with , N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { U_k } and { V_k } are i.i.d. random variable sequences with E [ U ₁ ] < ∞ and E [| V ₁ |] < ∞ , then there exists a function , r ≥ 0 and a function , r ≤ 0 almost surely such that for big t > 0, and and Proof . Let N′ ( t ) be the renewal random variables defined in Lemma 3. We consider that for 0 < l ₁ < 1/ pδ = < l ₂ , and r ≥ 0, where and Then, by the classical renewal theorem, lim _t→∞ N′ ( t )( ω )/ t = almost surely, we have for big t > 0, Nest, we consider that where η is a fuzzy variable with μ_η ( x ) = Π _R ( x ). Then, we have for big t > 0, for some ϵ > 0, and By similar argument, we have for big t > 0, for some ϵ > 0, and 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 { η_k } is μ _ξk(ω) ( x ) = Π _I ( x − U_k ( ω )) with Π _I (− a ) = 0, U_k ( ω ) ≥ a > 0 almost surely, and μ _ηk(ω) ( x ) = Π _R ( x − V_k ( ω )), for almost every ω ∈ Ω with , k = 1, 2, ⋯ , N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { U_k } and { V_k } are i.i.d. random variable sequences with E [ U ₁ ] < ∞ and E [| V ₁ |] < ∞ , then 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, and for almost all r ≤ 0, As a consequence, by Lemma 3.17 and the Lebesgue dominated convergence theorem, we have Theorem 3.19 ( [1] ). Let { X_n } be a sequence of random variables such that E | X_n | < ∞, n = 1, 2, 3,⋯ and X_n → X in probability. Then the following two propositions are equivalent: i) {| X_n |} is uniformly integrable ii)E | X_n | → 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 { η_k } is μ _ξk(ω) ( x ) = Π _I ( x − U_k ( ω )) with Π _I (− a ) = 0, U_k ( ω ) ≥ a > 0 almost surely, and μ _ηk(ω) ( x ) = Π _R ( x − V_k ( ω )) , for almost every ω ∈ Ω with , k = 1, 2, ⋯ , N ( t ) is the fuzzy random renewal variable, and C ( t ) is the total reward. If { U_k } and { V_k } are i.i.d. random variable sequences with E [ U ₁ ] < ∞ and E [| V ₁ |] < ∞ , then 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 be fuzzy random variables with μ _η′k(ω) ( x ) = Π _R′ ( x − V_k ( 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 to be fuzzy random variables with μ _{η′′k(ω)} ( x ) = Π _R′′ ( x − V_k ( ω )), for almost every ω ∈ Ω. Then we clearly have and It is also noted that for some positive real numbers v′ , v′′ > 0, respectively. Then we have Similarly, we have Then, since , k = 1, 2,⋯ is a sequence of pairs of i.i.d. random variables, by the classical stochastic renewal reward theorem, we have and Then, by Theorem 3.19, the class of random variables is uniformly integrable, and hence are uniformly integrable. Since { E [ C ( t )( ω )]/ t } is also uniformly integrable. Since, by Theorem 3.18, we have and hence by Theorem 3.19, again we have which completes the proof. □ Remark 3.7. If {( ξ_k , η_k ), 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] ). 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] ). 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