Some limit theorems on uncertain random sequences

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1 Journal of Intelligent & Fuzzy Systems 34 (218) DOI:1.3233/JIFS IOS Press 57 Some it theorems on uncertain random sequences Xiaosheng Wang a,, Dan Chen a, Hamed Ahmadzade b and Rong Gao c a School of Mathematics and Physics, Hebei University of Engineering, Handan, China b Department of Mathematical Sciences, University of Sistan and Baluchestan, Zahedan, Iran c School of Economics and Management, Hebei University of Technology, Tianjin, China Abstract. An uncertain random variable is a measurable function on the chance space. It is used to describe the mixing phenomena with both randomness and uncertainty. The uncertain random sequence is a sequence of uncertain random variables indexed by integers. Three types of convergence concept of uncertain random sequence have been defined, namely, convergence in distribution, convergence almost surely and convergence in measure, and some convergence theorems have been obtained. The main purpose of this paper is to provide some it theorems on uncertain random sequences. First, we construct two examples to illustrate the concepts of convergence almost surely and convergence in measure for a sequence of uncertain random variables. an inequality for uncertain random variable is presented, which states the relationship among chance measure, probability and uncertain measure. Several theorems about convergence of uncertain random sequences are obtained by Borel-Cantelli lemma which is given based on the properties of it superior. Finally, a convergence theorem for uncertain random series is established. The main results of this paper contain the relevant conclusions for random sequence and uncertain sequence. Keywords: Uncertain random variable, chance measure, it theorem 1. Introduction In probability theory, the it theorems on random sequence are one of the core research contents. The modes of convergence of random sequence are convergence in probability, convergence almost surely, convergence in distribution, complete convergence and so on. Besides, the types of random sequence are varied, such as independent random sequence, pairwise dependent random sequence, martingale and so on. So, it is important to find the necessary and sufficient condition or the sufficient condition for the convergence of different sequences under the different ways. Chow [4] proved some theorems about almost everywhere unconditional convergence Corresponding author. Xiaosheng Wang, School of Mathematics and Physics, Hebei University of Engineering, Handan 5638, China. Tel./Fax: ; xswang@ hebeu.edu.cn. of sums of independent identically distributed random variables. Heyde [7] gave a general sufficient condition for almost sure convergence to zero for normed and centered sums of independent random variables. Petrov [18] studied and summarized the convergence theorems for sums of independent random variables before 1975, including convergence weak, convergence complete, convergence in probability, convergence in distribution and convergence almost surely. Zanten [25] proved a theorem on the weak convergence of a properly normalized multivariate continuous local martingale. For recent developments, readers with an interest in convergence theorem may refer to Tang [2], Chen and Sung [2] and Lagodowski [11]. To depict fuzzy phenomena, Zadeh [23, 24] first proposed a concept of fuzzy set in 1965, and further defined possibility measure in In 23, Liu [14] presented four types of convergence concept of fuzzy /18/$ IOS Press and the authors. All rights reserved

2 58 X. Wang et al. / Some it theorems on uncertain random sequences sequence, namely, convergence almost surely, convergence in credibility, convergence in mean and convergence in distribution, and discussed the relationships among them. In order to describe the phenomena including both fuzziness and randomness, fuzzy random variable was first proposed by Kwakernaak [9, 1]. Subsequently, more and more scholars paid attention to this problem. Puri and Ralescu [19] gave a sufficient condition for almost everywhere convergence for a sequence of fuzzy random variables based on the definition of fuzzy martingale. To develop the approximation schemes to fuzzy random optimization problems, Liu et al. [17] introduced several convergence concepts for sequences of fuzzy random variables, such as convergence almost uniform, convergence in chance and convergence in distribution, then gave the necessary and sufficient conditions of convergence on different sequences and discussed the relations among various types of convergence. It is well known that probability theory and fuzzy set theory have been widely studied and applied. However, probability theory depends on the sample data, and fuzzy set theory does not emphasize the law of contradiction and the law of excluded middle. To deal with these two problems, uncertainty theory was established by Liu [12] in 27, and refined by Liu [15] in 21, which is based on normality, duality and subadditivity axioms. many researchers widely study the uncertainty theory and stay focused on the convergence theorem. You [22] defined the concept of convergence uniformly almost surely for uncertain sequence, and discussed the relations between convergence uniformly almost surely and convergence almost surely, convergence in measure, convergence in mean, convergence in distribution. Guo and Xu [6] gave a necessary and sufficient condition of convergence in mean square for an uncertain sequence. Chen et al. [3] introduced five convergence concepts of complex uncertain sequence, namely, convergence almost surely, convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely, and studied the relationships among them. For modelling complex systems with both randomness and uncertainty, chance theory was proposed by Liu [16] in 213. The convergence of uncertain random sequence is a tool in understanding the nature of uncertain random phenomena, big efforts have been put into studying convergence properties. The first it theorems for the law of large numbers of uncertain random variables were obtained by Yao and Gao [21]. Gao and Sheng [5] obtained a law of large numbers for uncertain random variables but not necessarily identically distributed. Ahmadzade et al. [1] verified the relationships between convergence almost surely and convergence in measure of uncertain random sequences. However, there are still a number of problems on it theorems. This paper will continue to study them. First, we will construct two examples to illustrate the concepts of convergence almost surely and convergence in measure for uncertain random sequences. we will study an inequality for an uncertain random variable which expounds the lower bound of the chance measure. Besides, we will establish Borel-Cantelli lemma for chance measure based on the properties of it superior. It is mentioned that, by invoking the continuity assumption of uncertain measure, this result in Ahmadzade et al. [1] was obtained. On the basis of this lemma, we will prove several convergence theorems for uncertain random sequences. They can provide a sufficient condition of convergence almost surely to zero for uncertain random sequences. At last, we will present and prove a theorem about convergence of uncertain random series. This theorem gives the condition to judge whether two uncertain random sequences have the same it properties. Also, some it theorems on random sequence and uncertain sequence are the corollaries of this paper. The rest of this paper is organized as follows. Section 2 will introduce some basic concepts about uncertainty theory and chance theory. an inequality for an uncertain random variable is investigated in Section 3. After that, some convergence theorems for uncertain random variables are studied in Section 4, and a theorem about convergence of uncertain random series is proved. Finally, some conclusions are made in Section Preinary In this section, we review some basic definitions and results about uncertainty theory and chance theory. Besides, we introduce two concepts of convergence Uncertainty theory Definition 1. (Liu [12]) Let L be a σ-algebra on a nonempty set Ɣ. Each element L is called an event. A set function M : L [, 1] is called an uncertain measure if it satisfies the following axioms:

3 X. Wang et al. / Some it theorems on uncertain random sequences 59 Axiom 1. (Normality Axiom) MƔ =1 for the universal set Ɣ. Axiom 2. (Duality Axiom) M +M c =1 for any event. Axiom 3. (Subadditivity Axiom) For every countable sequence of events 1, 2,,wehave M i M i. Note that the triplet (Ɣ, L, M) is called an uncertainty space. Besides, the product uncertain measure on the product σ-algebra L was defined by Liu [13] as follows. Axiom 4. (Product Axiom) Let (Ɣ k, L k, M k ) be uncertainty spaces for k = 1, 2, The product uncertain measure M is an uncertain measure satisfying M k = M k k k=1 k=1 where k are arbitrarily chosen events from L k for k = 1, 2,, respectively. Definition 2. (Liu [12]) An uncertain variable ξ is a measurable function from Ɣ to the set of real numbers R, i.e., for any Borel set B of real numbers, the set ξ B =γ Ɣ ξ(γ) B is an event. In order to describe an uncertain variable, a concept of uncertainty distribution is defined as follows. Definition 3. (Liu [12]) The uncertainty distribution of an uncertain variable ξ is defined by (x) = Mξ x for any real number x. In real life, we often consider a special kind of distribution, namely, regular uncertainty distribution. Regular distribution has a distinct feature that its inverse function exists. And its inverse function is called the inverse uncertainty distribution. That is, suppose ξ is an uncertain variable with regular uncertainty distribution (x). the inverse function 1 (α) is called the inverse uncertainty distribution of ξ. Inverse uncertainty distribution plays an important role in the operation of independent uncertain variables. Before introducing the operational law, a concept of independence of uncertain variables is presented in the following mathematical form. Definition 4. (Liu [13]) The uncertain variables ξ 1,ξ 2,,ξ n on the same uncertainty space are said to be independent if n n M (ξ i B i ) = M ξ i B i for any Borel sets B 1,B 2,,B n of real numbers. Theorem 1. (Liu [15]) Let ξ 1,ξ 2,,ξ n be independent uncertain variables with regular uncertainty distributions 1, 2,, n, respectively. If the function f (x 1,x 2,,x n ) is strictly increasing with respect to x 1,x 2,,x m and strictly decreasing with respect to x m+1,x m+2,,x n, then ξ = f (ξ 1,ξ 2,,ξ n ) is an uncertain variable with inverse uncertainty distribution 1 (α) = f ( 1 1 (α),, 1 m (α), 1 m Chance theory (1 α),, 1(1 α)). n Chance theory was initialized by Liu [16] in order to describe complex systems with not only uncertainty but also randomness. Let (Ɣ, L, M) be an uncertainty space and let (,A,Pr) be a probability space. the product (Ɣ, L, M) (,A,Pr) = (Ɣ,L A, M Pr) is called a chance space, in which Ɣ is the universal set, L A is the product σ-algebra and M Pr is the product measure. Definition 5. (Liu [16]) Let (Ɣ, L, M) (,A,Pr) be a chance space, and let L A be an uncertain random event. the chance measure of is defined as Ch = 1 xdx. Prω Mγ Ɣ (γ, ω) Liu [16] proved that the chance measure satisfies normality, duality and monotonicity properties, i.e.,

4 51 X. Wang et al. / Some it theorems on uncertain random sequences (1) ChƔ =1for the universal set Ɣ. (2) Ch +Ch c =1for any event. (3) Ch 1 Ch 2 for any events 1 2. Besides, Hou [8] verified the subadditivity of chance measure, i.e., Ch i Ch i for any countable sequence of events 1, 2, Theorem 2. (Liu [16]) Let (Ɣ, L, M) (,A,Pr) be a chance space. Ch A =M PrA for any L and any A A. Definition 6. (Liu [16]) An uncertain random variable ξ is a measurable function from a chance space (Ɣ, L, M) (,A,Pr) to the set of real numbers, i.e., ξ B =(γ, ω) ξ(γ, ω) B is an uncertain random event for any Borel set B of real numbers. An uncertain random variable ξ(γ, ω) degenerates to a random variable if it does not vary with γ. Thus a random variable is a special uncertain random variable. An uncertain random variable ξ(γ, ω) degenerates to an uncertain variable if it does not vary with ω. Thus an uncertain variable is a special uncertain random variable. Theorem 3. (Liu [16]) Let ξ be an uncertain random variable on the chance space (Ɣ, L, M) (,A,Pr), and let B be a Borel set of real numbers. ξ B is an uncertain random event with chance measure Chξ B = 1 Prω Mγ Ɣ ξ(γ, ω) B xdx. Definition 7. (Liu [16]) Let ξ be an uncertain random variable. its chance distribution is defined by for any x R. (x) = Chξ x 2.3. Two definitions of convergence This subsection introduces two concepts of convergence almost surely and convergence in measure for a sequence of uncertain random variables. Besides, we construct some examples to illustrate the concepts. Definition 8. (Ahmadzade [1]) The uncertain random sequence ξ n,n 1 is said to be convergent almost surely to the uncertain random variable ξ if there exists an event with Ch =1 such that ξ n(γ, ω) ξ(γ, ω) = for any (γ, ω). In that case we write ξ n ξ, a.s. Remark 1. When we do not emphasize (γ, ω), ξ n (γ, ω) ξ(γ, ω) = can be written as ξ n ξ =. Remark 2. The concept of convergence almost surely for uncertain random variables coincides with the concepts of convergence almost surely for random variables and for uncertain variables, respectively. Example 1. Take a chance space (Ɣ, L, M) (,A,Pr)tobeγ 1,γ 2, ω 1,ω 2, with Ch = (γ m,ω n ) 1 2 m+n. The uncertain random variables are defined by 2 i, if i = m = n ξ i (γ m,ω n ) =, otherwise for i = 1, 2, and ξ. ξ i converges a.s. to ξ. Definition 9. (Ahmadzade [1]) The uncertain random sequence ξ n,n 1 is said to be convergent in measure to the uncertain random variable ξ if Ch ξ n(γ, ω) ξ(γ, ω) ε = for any ε>. Example 2. Take a chance space (Ɣ, L, M) (,A,Pr) to be the interval [, 1] [, 1] with Borel algebra and Lebesgue measure. For any positive integer i, there is an integer j such that i = 2 j + k, where k is an integer between and 2 j 1. define uncertain random variables as 1, if both γ and ω are in ξ i (γ, ω) = [k/2 j, (k + 1)/2 j ], otherwise

5 X. Wang et al. / Some it theorems on uncertain random sequences 511 for i = 1, 2, and ξ. For any small number ε>, we have Ch ξ i ξ ε = 1 4 j as i. That is, the sequence ξ i converges in measure to ξ. Remark 3. Uncertain variables can be regarded as a special case of uncertain random variables. In the uncertain environment, convergence a.s. does not imply convergence in measure, and convergence in measure does not imply convergence a.s. Two examples were given by Liu [12]. 3. Inequality for uncertain random variable Inequality can be used to determine the upper and lower bounds. In this section, we present and prove an inequality for an uncertain random variable. Theorem 4. Let ξ be an uncertain random variable on the chance space (Ɣ, L, M) (,A,Pr). Ch ξ 2 k Mγ Ɣ ξ 2 k Prω ξ 2 k for real number k. Proof. By using Theorem 3, we conclude Ch ξ 2 k =Chξ [ 2 k, 2 k ] = 1 Prω Mγ Ɣ ξ(γ, ω) [ 2 k, 2 k ] xdx 1 ) Pr (ω ξ [ 2 k, 2 k ] = (ω Mγ Ɣ ξ [ 2 k, 2 k ] x)dx Mγ Ɣ ξ [ 2 k,2 k ] Prω ξ [ 2 k, 2 k ]dx = Mγ Ɣ ξ [ 2 k, 2 k ] Prω ξ [ 2 k, 2 k ] = Mγ Ɣ ξ 2 k Prω ξ 2 k. The proof is completed. Theorem 4 gives the lower bound of the chance measure. This inequality also can be used as a tool to prove some propositions. If uncertain random variable degenerates into random variable or uncertain variable, the corresponding results of the above theorem are as follows. Corollary 1. Let η be a random variable on the probability space (,A,Pr). for real number k. Ch η 2 k =Pr η 2 k Proof. By using Theorem 3, we conclude Ch η 2 k =ChƔ ( η 2 k ) = MƔ Pr η 2 k = Pr η 2 k. The proof is completed. Corollary 2. Let τ be an uncertain variable on the uncertainty space (Ɣ, L, M). for real number k. Ch τ 2 k =M τ 2 k Proof. By using Theorem 3, we conclude Ch τ 2 k =Ch( τ 2 k ) = M τ 2 k Pr = M τ 2 k. The proof is completed. 4. Some convergence theorems In this section, we introduce two concepts of it superior and it inferior for a sequence of uncertain random events, and establish a version of Borel- Cantelli lemma as Lemma 1. By using this lemma, we obtain some results about convergence of uncertain random sequences. Besides, we prove a convergence theorem for uncertain random series Convergence of uncertain random sequence This subsection introduces two concepts of it superior and it inferior for a sequence of uncertain random events, and establish a version of Borel- Cantelli lemma. On the basis of this lemma, we obtain

6 512 X. Wang et al. / Some it theorems on uncertain random sequences several theorems about convergence of uncertain random variables. Definition 1. Let 1, 2,, n, be a sequence of uncertain random events on the chance space (Ɣ, L, M) (,A,Pr). its it superior is defined by sup n =(γ, ω) (γ, ω) n for an infinite number of n. In other words, sup n is the set of all (γ, ω) which belongs to an infinite number of n. Definition 11. Let 1, 2,, n, be a sequence of uncertain random events on the chance space (Ɣ, L, M) (,A,Pr). its it inferior is defined by inf n =(γ, ω) (γ, ω) n for all but a finite number of n. In other words, inf n is the set of all (γ, ω) which belongs to all n, except for finite events. Example 3. Let n (n = 1, 2, 3, ) be a sequence of uncertain random events as follows. ( 2m+1 = (γ, ω) γ 2 + ω ) 2, m + 1 m =, 1, 2,, ( 2m = (γ, ω) γ 2 + ω ) 2, m m = 1, 2, 3,. sup n =(γ, ω) γ 2 + ω 2 < 4, inf n =(γ, ω) γ 2 + ω 2 1. The following Borel-Cantelli lemma has numerous applications in chance theory. Lemma 1. (Borel-Cantelli Lemma) Let n,n 1 be a sequence of uncertain random events on a chance space (Ɣ, L, M) (,A,Pr) such that Ch n <. Proof. Since k=n Ch sup n =. k k. Hence k=n Ch n < and sup n = Ch sup n = Ch k k=n Ch k The proof is completed. k=n Ch k. k=n Remark 4. In the random environment and uncertain environment, the above Borel-Cantelli lemma was still established. Based on the continuity assumption of uncertain measure, this result in Ahmadzade et al. [1] was obtained. We can now state our convergence theorem. Theorem 5. Let ξ n,n 1 be a sequence of uncertain random variables such that Ch ξ n >ε < for any ε>. ξ n = a.s. Proof. By hypothesis, for each k 1, Ch ξ n > 2 k <. Hence, by the Borel-Cantelli lemma, we get Ch sup( ξ n > 2 k ) =. Since the chance measure is self-dual, we have Ch inf n 2 k ) = 1. That is, for each k 1, ξ n 2 k for all n sufficiently large, except on a null event k. It follows that

7 X. Wang et al. / Some it theorems on uncertain random sequences 513 ξ n(γ, ω) = for all (γ, ω) / k. k=1 Since k=1 k is a null event, ξ n = a.s. follows. The proof is completed. The same thought can be used to discuss the convergence of uncertain random series ξ n and 1 n uncertain random sequences ξ k. Based on n k=1 this, we can further obtain the theoretical basis of common measurement methods under uncertain random environments. If uncertain random variables degenerate into random variables or uncertain variables, the corresponding results of the above theorem are as follows. Corollary 3. Let η n,n 1 be a sequence of random variables such that Pr η n >ε < for any ε>. η n = a.s. Proof. Random variables can be regarded as a special invoking Theorem 5 completes the proof. Corollary 4. Let τ n,n 1 be a sequence of uncertain variables such that M τ n >ε < for any ε>. τ n = a.s. Proof. Uncertain variables can be regarded as a special invoking Theorem 5 completes the proof. Theorem 6. Let ξ n,n 1 be a sequence of uncertain random variables. Suppose Ch ξ n >ε n < for positive constants ε n (n ). ξ n = a.s. Proof. By hypothesis, for each positive constants ε n, Ch ξ n >ε n <. Hence, by the Borel-Cantelli lemma, we get Ch sup( ξ n >ε n ) =. Since the chance measure is self-dual, we have Ch inf n ε n ) = 1. That is, for each positive constants ε n, ξ n ε n for all n sufficiently large, except on a null event k. It follows that n(γ, ω) = for all (γ, ω) / k. k=1 Since k=1 k is a null event, ξ n = a.s. follows. The proof is completed. Theorem 6 broadens the sufficient condition of Theorem 5. If uncertain random variables degenerate into random variables or uncertain variables, the corresponding results of the above theorem are as follows. Corollary 5. Let η n,n 1 be a sequence of random variables. Suppose Pr η n >ε n < for positive constants ε n (n ). η n = a.s. Proof. Random variables can be regarded as a special invoking Theorem 6 completes the proof. Corollary 6. Let τ n,n 1 be a sequence of uncertain variables. Suppose M τ n >ε n < for positive constants ε n (n ). τ n = a.s. Proof. Uncertain variables can be regarded as a special invoking Theorem 6 completes the proof.

8 514 X. Wang et al. / Some it theorems on uncertain random sequences 4.2. Convergence of uncertain random series When studying the it properties of uncertain random sequence, we can obtain some useful results with the aid of another known uncertain random sequence if they have the same it properties. So we need to give the condition to judge whether two uncertain random sequences have the same it properties. Theorem 7. Let ξ n,n 1 and ξ n,n 1 be two sequences of uncertain random variables such that Chξ n /= ξ n <. ξ n and ξ n converge almost surely together or diverge almost surely together. Proof. By the Borel-Cantelli lemma, we obtain Ch sup ξ n /= ξ n =. Since the chance measure is self-dual, we have Ch inf n = ξ n = 1. That is, for each (γ, ω) there is an integer N(γ, ω) such that n>n(γ, ω) implies ξ n (γ, ω) = ξ n (γ, ω), except on a null event k. Thus, ξ n and ξ n converge or diverge together, except on a null event k. i.e., ξ n and converge almost surely ξ n together or diverge almost surely together. The proof is completed. Theorem 7 gives the sufficient condition that two uncertain random series converge together or diverge together, except on a null event. When an uncertain random series ξ n converges, another unknown series ξ n also will converge, except on a null event. In addition, we can further study the it properties of uncertain random variables on the basis of this theorem. If uncertain random variables degenerate into random variables or uncertain variables, the corresponding results of the above theorem are as follows. Corollary 7. Let η n,n 1 and η n,n 1 be two sequences of random variables such that Prη n /= η n <. η n and η n converge almost surely together or diverge almost surely together. Proof. Random variables can be regarded as a special invoking Theorem 7 completes the proof. Corollary 8. Let τ n,n 1 and τ n,n 1 be two sequences of uncertain variables such that Mτ n /= τ n <. τ n and τ n converge almost surely together or diverge almost surely together. Proof. Uncertain variables can be regarded as a special invoking Theorem 7 completes the proof. 5. Conclusion The main contribution of this paper is providing some it theorems on uncertain random sequences. First, an inequality for an uncertain random variable has been studied. This inequality gives the lower bound of the chance measure, and also can be used as a tool to prove some propositions. Besides, for proving the convergence theorems we have proposed two concepts of it superior and it inferior for a sequence of uncertain random events, and given an example to show them. some it theorems for uncertain random variables have been established. Theorem 5 gives the sufficient condition of convergence almost surely to zero for uncertain random sequences. The same thought can be used to discuss the convergence of uncertain random series and 1 n uncertain random sequences ξ k, then we n k=1 can further obtain the theoretical basis of common measurement methods under uncertain random environments. In order to make the results in Theorem 5 more general and more convenient to apply, Theorem 6 broadens its sufficient condition. After that, a theorem about convergence of uncertain random series has been proved. This theorem provides the sufficient condition that two uncertain random series

9 X. Wang et al. / Some it theorems on uncertain random sequences 515 converge together or diverge together, except on a null event. When a known uncertain random series converges, another unknown series also will converge, except on a null event. Based on this theorem, we can further study the it properties of uncertain random variables. As special uncertain random variables, some it theorems on random variables and uncertain variables are the corollaries of this paper. Acknowledgments This work was supported by Hebei Natural Science Foundation (No. G ) and the Foundation of Hebei Education Department (ZD21716). References [1] H. Ahmadzade, Y. Sheng and M. Esfahani, On the convergence of uncertain random sequences, Fuzzy Optimization and Decision Making 16(2) (217), [2] P. Chen and S. Sung, On the strong convergence for weighted sums of negatively associated random variables, Statistics and Probability Letters 92 (214), [3] X. Chen, Y. Ning and X. Wang, Convergence of complex uncertain sequences, Journal of Intelligent and Fuzzy Systems 3(6) (216), [4] Y. Chow, Some convergence theorems for independent random variables, Annals of Mathematical Statistics 37(6) (1966), [5] R. Gao and Y. Sheng, Law of large numbers for uncertain random variables with different chance distributions, Journal of Intelligent and Fuzzy Systems 31(3) (216), [6] H. Guo and C. Xu, A necessary and sufficient condition of convergence in mean square for uncertain sequence, Information: An International Interdisciplinary Journal 16(2A) (213), [7] C. Heyde, On almost sure convergence for sums of independent random variables, Sankhyā: The Indian Journal of Statistics, Series A ( ) 3(4) (1968), [8] Y. Hou, Subadditivity of chance measure, Journal of Uncertainty Analysis and Applications 2 (214), Article 14. [9] H. Kwakernaak, Fuzzy random variables-i: Definitions and theorems, Information Sciences 15 (1978), [1] H. Kwakernaak, Fuzzy random variables-ii: Algorithms and examples for the discrete case, Information Sciences 17 (1979), [11] Z. Lagodowski, An approach to complete convergence theorems for dependent random fields via application of Fuk lcnagaev inequality, Journal of Mathematical Analysis and Applications 437(1) (216), [12] B. Liu, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 27. [13] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems 3(1) (29), 3 1. [14] B. Liu, Inequalities and convergence concepts of fuzzy and rough variables, Fuzzy Optimization and Decision Making 2(2) (23), [15] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 21. [16] Y. Liu, Uncertain random variables: A mixture of uncertainty and randomness, Soft Computing 17(4) (213), [17] Y. Liu, Z. Liu and J. Gao, The modes of convergence in the approximation of fuzzy random optimization problems, Soft Computing 13(2) (29), [18] V. Petrov, Sums of Independent Random Variables, Springer-Verlag, Berlin, [19] M. Puri and D. Ralescu, Convergence theorem for fuzzy martingales, Journal of Mathematical Analysis and Applications 16(1) (1991), [2] X. Tang, Strong convergence results for arrays of rowwise pairwise NQD random variables, Journal of Inequalities and Applications 213(1) (213), 1 8. [21] K. Yao and J. Gao, Law of large numbers for uncertain random variables, IEEE Transactions on Fuzzy Systems 24 (216), [22] C. You, On the convergence of uncertain sequences, Mathematical and Computer Modelling 49(3-4) (29), [23] L. Zadeh, Fuzzy sets, Information and Control 8 (1965), [24] L. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978), [25] H. Zanten, A multivariate central it theorem for continuous local martingales, Statistics and Probability Letters 5(3) (2),