On the convergence of fuzzy variables

1 Introduction

The world is neither random nor fuzzy, but sometimes it can be analyzed by probability theory, sometimes by credibility theory. As a branch of mathematics, probability theory studies the behavior of random phenomena. Attempt to deal with non-random phenomena, Zadeh [30] introduced the concept of fuzzy set via membership function. Furthermore, Zadeh [31] proposed the concept of possibility measure for measuring a fuzzy event. During the past years, many researchers contributed a lot to this area. Based on fuzzy set theory, kinds of convergence for fuzzy numbers were discussed by researchers. For example, Altin, Et and Colak [2] introduced the concepts of lacunary statistical convergence and lacunary strongly convergence of generalized difference sequences for fuzzy numbers in 2006. After that, the concepts of statistical convergence of order β and strong p-Cesàro summability of order β for sequences of fuzzy numbers were introduced by Altinok, Altin and Isik [1]. Then the concepts of δ λ m -statistical convergence of order β and strong δ λ p m -summability of order β for sequences of fuzzy numbers were presented by Karakas, Altin and Altinok [6].

As we all know, self-duality is important and necessary both in theory and application. However, possibility measure has no self-duality. Thus, in 2002, Liu and Liu [11] proposed a self-duality measure, credibility measure. Liu [14] gave an axiomatic foundation for credibility theory and a survey of credibility theory was given by Liu [15] in 2006. Since then, credibility theory has been developed steadily. Considering sequence convergence plays an extremely important role in credibility theory, Liu [13] gave four types of convergence concept for fuzzy variables: convergence almost surely, convergence in credibility, convergence in mean, and convergence in distribution in 2003. Besides, based on credibility theory, some convergence properties of credibility distribution for fuzzy variables were discussed by Jiang [5] and Ma [20]. In order to describe the phenomena involving fuzziness and randomness simultaneously, Kwakernaak [7] proposed the concept of fuzzy random variables. After that, researchers such as Puri and Ralescu [21], Kruse and Meyer [8], Liu and Liu [17] gave kinds of definitions of fuzzy random variables according to different requirements of measurability. For fuzzy random variable, a chance measure was given by Gao and Liu [3]. In addition, random fuzzy variable and the corresponding chance measure were introduced by Liu [12]. Liu, Liu and Gao [19] proved the convergence theorems for sequences of integrable fuzzy random variables, such as dominated convergence theorem and bounded convergence theorem. In addition, as an extension of convergence for fuzzy variables, some definitions of sequences convergence for uncertain variables were proposed by Liu [16] and You [24]. At the same time, some mathematical properties of those convergence were also given. Then the Cauchy convergence of uncertain variables and the sufficient conditions of convergence almost surely were presented by Xia [23]. After that, Gao [4] discussed the convergence theorems for expected value of uncertain variables. Furthermore, r-order convergence and dual convergence of uncertain variable sequences were researched by Yuan [27], Yuan, Zhu and Guo [28]. Then Chen, Ning and Wang [29] introduced the convergence concepts of complex uncertain variables. In addition, You and Yan [26] gave a new concept of sequence convergence for uncertain variables that is convergent in p-distance and the relationships among concepts of convergence for uncertain variables sequences were studied by You and Yan [25].

In fact, many researchers discussed convergence concepts in classical measure theory, probability theory, credibility theory and chance theory, and the relationships between them. The interested reader may consult Liu [13], Wang and Liu [22], Li and Liu [9] and Liu [16], and books related to measure theory. Inspired by these, in this paper, a further investigation into the mathematical properties of convergence for fuzzy variables will be made.

The rest of the paper is organized as follows. Section 2 will recall some definitions and theorems in credibility theory. In Section 3, some convergence theorems for fuzzy variables sequence will be given. Finally, a brief conclusions will be obtained in Section 4.

2 Preliminaries

Let T be a nonempty set, and P the power set of Θ. Each element in P is called an event. To measure a fuzzy event, credibility measure Cr was introduced as a set function satisfying the following axioms:

Axiom 1. (Normality) Cr {Θ} =1.

Definition 2.2. (Liu and Liu [11]) Let ξ be a fuzzy variable. Then the expected value of ξ is defined by E [ ξ ] = ∫ 0 + ∞ Cr { ξ ≥ r } d r - ∫ - ∞ 0 Cr { ξ ≤ r } d r provided that at least one of the two integrals is finite.

Definition 2.3. (Liu [13]) Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables defined on credibility space ( Θ , P , Cr ) . The sequence {ξ _n } is said to be convergent a.s. to ξ if and only if there exists A with Cr {A} =1 such that lim n → ∞ ∣ ξ n ( θ ) - ξ ( θ ) ∣ = 0 for every θ ∈ A.

Definition 2.4. (Liu [13]) Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables defined on credibility ( Θ , P , Cr ) . We say that the sequence {ξ _n } converges in credibility to ξ if lim n → ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ } = 0 for every ɛ > 0.

Definition 2.5. (Liu [13]) Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables with finite expected values defined on ( Θ , P , Cr ) . We say that the sequence {ξ _n } converges in mean to ξ if lim n → ∞ E [ ∣ ξ n - ξ ∣ ] = 0 .

Theorem 2.1. (Wang and Liu [22]) Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables. If the sequence {ξ _n } convergence in credibility to ξ, then {ξ _n } converges a.s. to ξ.

Theorem 2.2. (Liu [13]) Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables. If the sequence {ξ _n } converges in mean to ξ, then {ξ _n } converges in credibility to ξ.

A fuzzy variable ξ is said to be essentially bounded if there exists a positive number M such that Cr { ξ ≤ - M } = 0 and Cr { ξ ≤ M } = 1 .

A sequence {ξ _k } of fuzzy variables is said to be uniformly essentially bounded if there is a positive number M such that for each k, we have Cr { ξ k ≤ - M } = 0 and Cr { ξ k ≤ M } = 1 .

Theorem 2.3. (Bounded Convergence Theorem, Liu and Wang [18]) Suppose that {ξ _n } is a sequence of uniformly essentially bounded fuzzy variables. If {ξ _n } converges in credibility to ξ, then lim n → ∞ E [ ξ n ] = E [ ξ ] .

Theorem 2.4. (Lin [10]) Let f: R → R be a convex function. Then there exists a number L > 0 such that ∣ f ( x 1 ) - f ( x 2 ) ∣ ≤ L ∣ x 1 - x 2 ∣ , for any x₁, x₂ ∈ R.

3 Some convergence properties for fuzzy variables

Theorem 3.1. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables. Then {ξ _n } converges in credibility to ξ if ∑ n = 1 ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ } < ∞ , for any ɛ > 0.

Proof. If ∑ n = 1 ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ } < ∞ , then lim n → ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ } = 0 . That is to say, {ξ _n } converges in credibility to ξ.

Definition 3.1. Suppose that ξ₁, ξ₂, ⋯ are fuzzy variables. For any ɛ > 0, if there exists an event A with Cr {A} =1 and N ∈ N⁺ such that for every θ ∈ A, we have ∣ ξ n ( θ ) - ξ m ( θ ) ∣ < ɛ , for any n, m > N, then the sequence {ξ _n } is called a Cauchy sequence a.s.

Example 3.1. Take ( Θ , P , Cr ) to be {θ₁, θ₂, ⋯} with Cr { θ j } = 1 2 for j = 1, 2, ⋯. The fuzzy variables are defined by ξ k ( θ j ) = { 1 j , if k = j 0 , otherwise . For any ɛ > 0, taking A = Θ and N = [ 1 ɛ ] + 1 , we have ∣ ξ n ( θ ) - ξ m ( θ ) ∣ < 1 N < ɛ , for any n, m > N and every θ _j ∈ A. Therefore, the sequence {ξ _k } is a Cauchy sequence a.s.

Theorem 3.2. Suppose that ξ, ξ₁, ξ₂, ⋯ are fuzzy variables. Then the sequence {ξ _n } converges a.s. to ξ if and only if {ξ _n } is a Cauchy sequence a.s.

Proof. If {ξ _n } converges a.s. to ξ, then there exists a fuzzy event A with Cr {A} =1 such that for any ɛ > 0, there exists N ∈ N⁺, we have ∣ ξ n ( θ ) - ξ ( θ ) ∣ < ɛ 2 , and ∣ ξ m ( θ ) - ξ ( θ ) ∣ < ɛ 2 , when n, m > N, for every θ ∈ A. Thus, ∣ ξ n ( θ ) - ξ m ( θ ) ∣ < & ∣ ξ n ( θ ) - ξ ( θ ) ∣ + ∣ ξ m ( θ ) - ξ ( θ ) ∣ < & ɛ . That is, {ξ _n } is a Cauchy sequence a.s.

On the contrary, if {ξ _n } is a Cauchy sequence a.s., then for any ɛ > 0, there exists N₁ ∈ N⁺ such that ∣ ξ n ( θ ) - ξ N 1 ( θ ) ∣ < ɛ , when n > N₁. If {ξ _n } does not converge a.s., then there exists θ^* ∈ A and ɛ₀ > 0, for any N₂ ∈ N⁺, we have ∣ ξ N 3 ( θ * ) - ξ ( θ * ) ∣ > ɛ 0 , when N₃ > N₂. Let M = ξ N 3 ( θ * ) + ξ N 1 ( θ * ) 2 . Then ξ N 1 ( θ * ) = 2 M - ξ N 3 ( θ * ) . Thus ∣ ξ N 3 ( θ * ) - ξ N 1 ( θ ) ∣ = & ∣ ξ N 3 ( θ * ) - 2 M + ξ N 3 ( θ * ) ∣ = & 2 ∣ ξ N 3 ( θ * ) - M ∣ > 2 ɛ 0 , when N₂ > N₁. Thus {ξ _n } is not a Cauchy sequence a.s. A contradiction proves the theorem.

Therefore, {ξ _n } converges a.s. to ξ.

Definition 3.2. Suppose ξ₁, ξ₂, ⋯ are fuzzy variables. We say that the sequence {ξ _n } is a Cauchy sequence in credibility, if for any ɛ > 0, δ > 0, there exists N such that Cr { ∣ ξ n - ξ m ∣ > ɛ } < δ , when n, m > N.

Example 3.2. Take ( Θ , P , Cr ) to be {θ₁, θ₂, ⋯} with Cr { θ 1 } = 1 2 and Cr { θ j } = 1 j , for j = 2, 3, ⋯. The fuzzy variables are defined by ξ k ( θ j ) = { j , if k = j 0 , otherwise . For any δ > 0, taking 0 < ɛ < 1 and N = [ 2 δ ] + 1 , we have Cr { ∣ ξ n - ξ m ∣ > ɛ } = 1 n + 1 m < 2 N < δ , for any n, m > N. Therefore, the sequence {ξ _k } is a Cauchy sequence in credibility.

Theorem 3.3. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables. If {ξ _n } is a Cauchy sequence in credibility, then the sequence {ξ _n } converges a.s. to ξ.

Proof. If {ξ _n } is a Cauchy sequence in credibility, then for any ɛ > 0, δ > 0, there exists N such that Cr { ∣ ξ n - ξ m ∣ > ɛ } < δ , when n, m > N. Furthermore, we can obtain lim n , m → ∞ Cr { ∣ ξ n - ξ m ∣ ≥ ɛ } = 0 . If {ξ _n } does not converge a.s. to ξ, then there exist an element θ^∗ ∈ Θ with Cr {θ^∗} >0 such that ξ _n (θ^∗) ↛ ξ (θ^∗) as n → ∞. In other words, there exist a small number ɛ > 0 and subsequences ξ _nk (θ^∗) and ξ _mk (θ^∗) of ξ _n (θ^∗) such that ∣ ξ nk ( θ ∗ ) - ξ mk ( θ ∗ ) ∣ ≥ ɛ for any k. It follows from Axiom 2 that Cr { ξ nk - ξ mk ∣ ≥ ɛ } ≥ Cr { θ ∗ } > 0 for any k.

Thus {ξ _n } is not a Cauchy sequence in credibility. A contradiction proves the theorem.

Theorem 3.4. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables. If {ξ _n } converges in credibility to ξ, then {ξ _n } is a Cauchy sequence in credibility.

Proof. If {ξ} converges in credibility to ξ, then lim n → ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ } = 0 . That is to say, for any δ > 0, there exists N such that Cr { ∣ ξ n - ξ ∣ > ɛ } < δ 2 , Cr { ∣ ξ m - ξ ∣ > ɛ } < δ 2 , for any n, m > N.

Since Cr { ∣ ξ n - ξ m ∣ > ɛ } ≤ & Cr { ∣ ξ n - ξ ∣ > ɛ 2 } + Cr { ∣ ξ - ξ m ∣ > ɛ 2 } < & δ 2 + δ 2 = δ , we conclude that {ξ _n } is a Cauchy sequence in credibility.

Theorem 3.5. Let ξ, ξ₁, ξ₂, ⋯ be fuzzy variables. If there exists a sequence of numbers {ɛ _n } such that ∑ n = 1 ∞ ɛ n < + ∞ and ∑ n = 1 ∞ Cr { ∣ ξ n + 1 - ξ n ∣ > ɛ n } < + ∞ , then {ξ _n } converges a.s. to ξ.

Proof. Since ∑ n = 1 ∞ Cr { ∣ ξ n + 1 - ξ n ∣ > ɛ n } < + ∞ , we have lim n → ∞ Cr { ∣ ξ n + 1 - ξ n ∣ > ɛ n } = 0 . From Cr { ⋂ m = 1 ∞ ⋃ n = m ∞ { ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ > ɛ n } } ≤ lim m → ∞ Cr { ⋃ n = m ∞ { ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ > ɛ n } } ≤ lim m → ∞ ∑ n = m ∞ Cr { ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ > ɛ n } = lim n → ∞ Cr { ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ > ɛ n } , we have Cr { ⋂ m = 1 ∞ ⋃ n = m ∞ { ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ > ɛ n } } = 0 , therefore Cr { ⋃ m = 1 ∞ ⋂ n = m ∞ { ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ ≤ ɛ n } } = 1 . Let T = ⋃ m = 1 ∞ ⋂ n = m ∞ { θ ∈ Θ | ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ ≤ ɛ n } . Since ∑ n = 1 ∞ ɛ n < + ∞ , for any ɛ > 0, there exists M₀ such that ɛ M 0 + ɛ M 0 + 1 + ɛ M 0 + 2 + ⋯ = ∑ n = M 0 ∞ ɛ n ≤ ɛ . If θ ∈ T, there exists m₀ such that θ ∈ ⋂ n = m 0 ∞ { θ ∈ Θ | ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ ≤ ɛ n } . Taking M ≥ max {m₀, M₀}, we have ɛ M + ɛ M + 1 + ⋯ < ɛ and θ ∈ ⋂ n = M ∞ { θ ∈ Θ | ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ ≤ ɛ n } . Denote S = ⋂ n = m ∞ { θ ∈ Θ | ∣ ξ n + 1 ( θ ) - ξ n ( θ ) ∣ ≤ ɛ n } , then T ⊆ S. Therefore, Cr {S} =1. That is, as long as n ≥ M, we have ∣ξ_n+1 - ξ _n ∣ ≤ ɛ _n , for any θ ∈ S, then ∣ ξ M - ξ M + n ∣ ≤ ∣ ξ M - ξ M + 1 ∣ + ⋯ + ∣ ξ M + n - 1 - ξ M + n ∣ ≤ ɛ M + ⋯ + ɛ M + n - 1 ≤ ɛ . Thus {ξ _n } is a Cauchy sequence a.s. It follows from Theorem 3.2 that {ξ _n } converges a.s. to ξ.

Theorem 3.6. Let ξ, ξ₁, ξ₂, ⋯ be fuzzy variables and let f: R → R be a convex function. If {ξ _n } converges a.s. to ξ, then {f (ξ _n )} converges a.s. to f (ξ).

Proof. Since f is a convex function, it follows from Theorem 2.4 that there exists a constant k such that ∣ f ( x ) - f ( y ) ∣ ≤ k ∣ x - y ∣ , for any x, y ∈ R. Replacing x with ξ _n and y with ξ, we obtain ∣ f ( ξ n ) - f ( ξ ) ∣ ≤ k ∣ ξ n - ξ ∣ . If {ξ _n } converges a.s. to ξ, then for any ɛ > 0, there exists an event A with Cr {A} =1 such that for every θ ∈ A ∣ ξ n - ξ ∣ < ɛ k . Then ∣ f ( ξ n ) - f ( ξ ) ∣ ≤ k ∣ ξ n - ξ ∣ < k · ɛ k = ɛ , i.e. lim n → ∞ ∣ f ( ξ n ) - f ( ξ ) ∣ = 0 .

That is, {f (ξ _n )} converges a.s. to f (ξ).

Theorem 3.7. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables. If {ξ _n } converges to ξ in credibility and f: R → R is a convex function, then {f (ξ _n )} converges to f (ξ) in credibility.

Proof. Since f is a convex function, it follows from Theorem 2.4 that there exists a constant k such that ∣ f ( x ) - f ( y ) ∣ ≤ k ∣ x - y ∣ , for any x, y ∈ R. Since {ξ _n } converges to ξ in credibility, we have lim n → ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ k } = 0 , for every ɛ > 0. Thus lim n → ∞ Cr { ∣ ξ n - ξ ∣ < ɛ k } = 1 . Then ∣ f ( ξ n ) - f ( ξ ) ∣ ≤ k ∣ ξ n - ξ ∣ < k · ɛ k = ɛ . Therefore lim n → ∞ Cr { ∣ f ( ξ n ) - f ( ξ ) ∣ < ɛ } = 1 , thus lim n → ∞ Cr { ∣ f ( ξ n ) - f ( ξ ) ∣ ≥ ɛ } = 0 . That is to say, {f (ξ _n )} converges to f (ξ) in credibility.

Theorem 3.8. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables, and f: R → R is a convex function. If {ξ _n } converges in mean to ξ, then {f (ξ _n )} converges in mean to f (ξ).

Proof. If {ξ _n } converges in mean to ξ, then lim n → ∞ E [ ∣ ξ n - ξ ∣ ] = 0 . By using Theorem 2.2, for any ɛ > 0, lim n → ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ } = 0 .

Since f is a convex function, it follows from Theorem 3.7 that lim n → ∞ Cr { ∣ f ( ξ n ) - f ( ξ ) ∣ ≥ ɛ } = 0 .

At the same time, we can infer that ∣f (ξ _n ) - f (ξ)∣ is bounded. That is ∣f (ξ _n ) - f (ξ)∣ is uniformly essentially bounded. According to Theorem 2.3, we have lim n → ∞ E [ ∣ f ( ξ n ) - f ( ξ ) ∣ ] = 0 . Therefore, {f (ξ _n )} converges in mean to f (ξ).

Corollary 3.1. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables, and f: R → R is a convex function. If {ξ _n } converges in credibility to ξ, then {f (ξ _n )} converges a.s. to f (ξ).

Proof. Note that the result can be derived from Theorem 3.7 and Theorem 2.1.

Corollary 3.2. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables, and f: R → R is a convex function. If {ξ _n } converges in mean to ξ, then {f (ξ _n )} converges in credibility to f (ξ).

Proof. Note that the result can be derived from Theorem 3.8 and Theorem 2.2.

Theorem 3.9. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables, and f: R → R is a continuous function. If {ξ _n } converges a.s. to ξ, then {f (ξ _n )} converges a.s. to f (ξ).

Proof. Since f is a continuous function, for every ɛ > 0, there exists δ > 0 such that ∣ξ _n - ξ ∣ < δ implies ∣f (ξ _n ) - f (ξ) ∣ < ɛ. Therefore, ∣f (ξ _n ) - f (ξ) ∣ ≥ ɛ implies ∣ξ _n - ξ ∣ ≥ δ. Thus { ∣ f ( ξ n ) - f ( ξ ) ∣ ≥ ɛ } ⊂ { ∣ ξ n - ξ ∣ ≥ δ } . If {ξ _n } converges a.s. to ξ, then for any δ > 0, there exists an event A with Cr {A} =1 such that for every θ ∈ A, ∣ξ _n - ξ ∣ < δ, i.e. Cr { ∣ ξ n - ξ ∣ ≥ δ } = 0 , thus Cr { ∣ f ( ξ n ) - f ( ξ ) ∣ ≥ ɛ } = 0 . Therefore, for any ɛ > 0 and θ ∈ A, ∣ f ( ξ n ) - f ( ξ ) ∣ < ɛ . That is, {f (ξ _n )} converges a.s. to f (ξ).

Theorem 3.10. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables. If {ξ _n } converges to ξ in credibility and f: R → R is a continuous function, then {f (ξ _n )} converges to f (ξ) in credibility.

Proof. If {ξ _n } converges to ξ in credibility, then for every ɛ > 0, lim n → ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ } = 0 . Since f is a continuous function, for every ɛ > 0, there exists δ > 0 such that ∣ξ _n - ξ ∣ < δ implies ∣f (ξ _n ) - f (ξ) ∣ < ɛ. Therefore, ∣f (ξ _n ) - f (ξ) ∣ ≥ ɛ implies ∣ξ _n - ξ ∣ ≥ δ. Thus Cr { ∣ f ( ξ n ) - f ( ξ ) ∣ ≥ ɛ } ≤ Cr { ∣ ξ n - ξ ∣ ≥ δ } , taking the limitation of n→ ∞ on both sides of above inequality, we have lim n → ∞ Cr { ∣ f ( ξ n ) - f ( ξ ) ∣ ≥ ɛ } = 0 . That is to say, {f (ξ _n )} converges to f (ξ) in credibility.

Theorem 3.11. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables, and f: R → R is a continuous function. If {ξ _n } converges in mean to ξ, then {f (ξ _n )} converges in mean to f (ξ).

Proof. If {ξ _n } converges in mean to ξ, then lim n → ∞ E [ ∣ ξ n - ξ ∣ ] = 0 . By using Theorem 2.2, for any ɛ > 0, lim n → ∞ Cr { ∣ ξ n - ξ ∣ ≥ ɛ } = 0 .

Since f is a continuous function, it follows from Theorem 3.10 that lim n → ∞ Cr { ∣ f ( ξ n ) - f ( ξ ) ∣ ≥ ɛ } = 0 .

At the same time, we can infer that ∣f (ξ _n ) - f (ξ)∣ is uniformly essentially bounded. According to Theorem 2.3, we have lim n → ∞ E [ ∣ f ( ξ n ) - f ( ξ ) ∣ ] = 0 . Therefore, {f (ξ _n )} converges in mean to f (ξ).

Corollary 3.3. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables, and f: R → R is a continuous function. If {ξ _n } converges in credibility to ξ, then {f (ξ _n )} converges a.s. to f (ξ).

Proof. Note that the result can be derived from Theorem 3.10 and Theorem 2.1.

Corollary 3.4. Suppose ξ, ξ₁, ξ₂, ⋯ are fuzzy variables, and f: R → R is a continuous function. If {ξ _n } converges in mean to ξ, then {f (ξ _n )} converges in credibility to f (ξ).

Proof. Note that the result can be derived from Theorem 3.11 and Theorem 2.2.

4 Conclusions

This paper mainly devoted to further investigating into the convergence theory of fuzzy variable sequences. To begin with, Cauchy sequence a.s. and Cauchy sequence in credibility were defined in this paper. Next, for fuzzy variable sequences, convergence conditions such as Cauchy convergence a.s., Cauchy convergence in credibility, convergence almost surely, convergence in credibility and convergence in mean were given.