On the convergence of uncertain random sequences

Fuzzy Optim Decis Making (217) 16:25 22 DOI 1.17/s17-16-9242-z On the convergence of uncertain random sequences H. Ahmadzade 1 Y. Sheng 2 M. Esfahani 3 Published online: 4 June 216 Springer Science+Business Media New York 216 Abstract In this paper, a useful inequality for central moment of uncertain random variables is proved. Based on this inequality, a convergence theorem for sum of uncertain random variables is derived. A Borel Cantelli lemma for chance measure is obtained based on the continuity assumption of uncertain measure. Finally, several convergence theorems for uncertain random sequences are established. Keywords Chance theory Uncertain random variable Chance distribution Moments Convergence 1 Introduction Most human decisions are made in the presence of an uncertain environment. The performance of a specific uncertainty can be represented by a particular type of mathematical measure. Besides, randomness is a practical model and a probability measure is a normalized additive measure where rigorous mathematical foundations of probability theory were given by Kolmogorov (1933). Prior to today, probability theory and fuzzy set theory are two common mathematical tools to model indeterminacy phenomena and have been widely applied in information theory, engineering, management science, and so on. We know a fundamental premise of applying probability theory is that the estimated probability B Y. Sheng shengyuhong1@sina.com 1 Department of Mathematical Sciences, University of Sistan and Baluchestan, Zahedan, Iran 2 College of Mathematical and System Sciences, Xinjiang University, Ürümqi 8346, China 3 Department of Mathematics, Velayat University, Iranshahr, Iran

26 H. Ahmadzade et al. distribution function is close enough to the real frequency. However, in real life, natural language statements like strength of bridge, about one million tons, tall, and most, due to lack of observed data and the complexity of environment, are typically used to express imprecise information or knowledge. In this case, information and knowledge cannot be described well by random variables. For fuzzy set theory, it is still challenged by many scholars after it was founded. Liu (27) presented several paradoxes to show that fuzzy variable and fuzzy set are not suitable for modelling uncertain quantities and unsharp concepts. Therefore, we review several researches on topics related to probability and uncertainty theories. Probability theory is a branch of mathematics for studying the behavior of random phenomena. In order to deal with randomness, Kolmogorov (1933) defined a probability measure as a set function satisfying normality, nonnegativity, and additivity axioms. Before applying it in practice, we should first obtain the probability distribution via statistics, or test of the probability distribution to make sure it is close enough to the real frequency, either of which is based on a lot of observed data. However, due to the technological or economical difficulties, we have no observed data sometimes. In this case, we have to rely on domain experts evaluating their belief degrees about the chances that the possible events happen. As a result, the probability theory is not applicable in this case, otherwise it may lead to counterintuitive results. Liu (212) gave some examples of it. In order to deal with indeterminate phenomena, Liu (27) established uncertainty theory as a branch of axiomatic mathematics for modelling belief degrees. On the other hand, Liu (27) first presented an uncertain measure as a set function satisfying normality, duality, subadditivity, and product axioms. As a fundamental concept in uncertainty theory, the uncertain variable was presented by Liu (27). In order to describe uncertain variable, Liu (27) introduced the concept of uncertainty distribution. Liu (21) proposed the concept of inverse uncertainty distribution, and Liu (213) verified a sufficient and necessary condition for it. After that, many researchers widely studied the uncertainty theory and made significative progress. Gao (29) studied the properties of continuous uncertain measure. Furthermore, a measure inversion theorem was given by Liu (21) that may yield uncertain measures of some special events from the uncertainty distribution of the corresponding uncertain variable. In addition, the concept of independence was proposed by Liu (29). After,Liu (21) presented the operational law of uncertain variables. In order to rank uncertain variables, Liu (27) proposed the concept of expected value of uncertain variables. The linearity of expected value operator was verified by Liu (21). Meanwhile, Liu (27) presented the concept of variance of uncertain variables. Furthermore, several authors proposed some formulas to calculate the variance through uncertainty distribution, for more details, see Liu (215). Yao (215) proposed a formula to calculate the variance using inverse uncertainty distribution. Liu (27) introduced the concept of moments and provided a formula to calculate the moments via uncertainty distribution. However, in many cases, randomness and uncertainty exist simultaneously in a complex system. Inspired by Kwakernaak (1978) and Puri and Ralescu (1986) uncertain random variable was first defined by Liu (213a) to describe complex systems in which uncertainty and randomness frequently appear together. Thus, in order to describe such a system, Liu (213a) first proposed chance theory, which is a mathematical

On the convergence of uncertain random sequences 27 methodology for modelling complex systems with both uncertainty and randomness, including chance measure, uncertain random variable, chance distribution, expected value, variance and so on. Following that, Liu (213b) presented the operational law of uncertain random variable, the formula of expected value and proposed uncertain random programming as a branch of mathematical programming involving uncertain random variables. As an important branch of chance theory, Gao and Yao (215) proposed an uncertain random process to model a dynamic uncertain random system. Furthermore, as an extension, Zhou et al. (214) introduced uncertain random multi-objective programming. In order to deal with uncertain random systems, Liu and Ralescu (214) invented the tool of uncertain random risk analysis. As everyone knows, the variance of uncertain random variable provides a degree of the spread of the distribution around its expected value. Guo and Wang (214) proved a formula for calculating the variance of uncertain random variables based on uncertainty distribution. Sheng and Yao (214) proposed a formula to calculate the variance using chance distribution and inverse chance distribution. In addition, Liu (214) presented a concept of uncertain random network and obtained the shortest path distribution of an uncertain random network. Sheng and Gao (214) obtained the maximum flow of chance distribution of an uncertain random network. Since sequence convergence plays an important role in chance theory, big efforts have been put into studying convergence properties. Yao and Gao (215) verified a law of large numbers for uncertain random variables. In this paper, we will first study a useful inequality for central moment of uncertain random variables. Then we will prove the Borel Cantelli lemma for chance measure based on the continuity assumption of uncertain measure. By invoking main results, we will establish several convergence theorems for uncertain random variables. The rest of this paper is organized as follows. Section 2 presents some basic concepts and theorems about chance theory including some concepts of uncertain random variables, expected value, variance and so on. An inequality for central moments of uncertain random variable is proved in Sect. 3. Also, some convergence theorems for uncertain random variables are obtained in Sect. 4. Finally, some conclusions are given in Sect. 5. 2 Preliminaries In this section, we review some concepts in uncertainty theory and chance theory, including uncertain variable, chance distribution, operational law, expected value, variance and so on. 2.1 Uncertain variables In this subsection, we recall several concepts of uncertainty theory that will be used in the next sections. For more details, see Liu (27, 29). Let L be a σ -algebra on a nonempty set Ɣ.AsetfunctionM : L [, 1] is called an uncertain measure if it satisfies the following axioms:

28 H. Ahmadzade et al. (i) (Normality) MƔ =1 for the universal set Ɣ. (ii) (Duality) M +M c =1 for any event. (iii) (Subadditivity) For every countable sequence of events 1, 2,..., we have M i M i. (iv) (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=1 k = k=1 M k k where k are arbitrarily chosen events from L k for k = 1, 2,..., respectively. Definition 1 (Liu 27) An uncertain variable ξ is a function from an uncertainty space (Ɣ, L, M) to the set of real numbers such that ξ B is an event for any Borel set B. Definition 2 (Liu 27) The uncertain variables ξ 1,ξ 2,...,ξ n 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. Theorem 1 (Liu 27) Let ξ 1,ξ 2,...,ξ n be independent uncertain variables, and f 1, f 2,..., f n be measurable functions. Then f 1 (ξ 1 ), f 2 (ξ 2 ),..., f n (ξ n ) are independent uncertain variables. Definition 3 (Liu 27) The events 1, 2,..., n are said to be independent if M n i = n M i such that i are arbitrarily chosen from i, i c,ɣ, i = 1, 2,...,n, respectively, where Ɣ is sure event. Definition 4 (Liu 29) The uncertainty distribution of an uncertain variable ξ is defined by for any real number x. (x) = Mξ x Definition 5 (Liu 29) Let ξ be an uncertain variable with regular uncertainty distribution (x). Then the inverse function 1 (α) is called the inverse uncertainty distribution of ξ. Theorem 2 (Liu 29) Let ξ 1,...,ξ n be independent uncertain variables with regular uncertainty distributions 1, 2,..., n, respectively. If f is a strictly increasing

On the convergence of uncertain random sequences 29 function, then ξ = f (ξ 1,ξ 2,...,ξ n ) is an uncertain variable with inverse uncertainty distribution 1 (α) = f ( 1 1 (α),..., 1 (α)). In order to obtain main results of convergence of uncertain random variables, we review the following concept. Definition 6 (Gao 29) An uncertain measure M is called continuous if for any sequence of events i with i,wehavem lim i i = lim i M i. n 2.2 Uncertain random variable The chance space refers to the product (Ɣ, L, M) (, A, Pr), in which (Ɣ, L, M) is an uncertainty space and (, A, Pr) is a probability space. Definition 7 (Liu 213a) Let(Ɣ, L, M) (, A, Pr) be a chance space, and L A be an uncertain random event. Then the chance measure of is defined as Ch = 1 Prω Mγ Ɣ (γ, ω) rdr. It is mentioned that a chance measure satisfies normality, duality, and monotonicity properties, that is (i) ChƔ =1; (ii) Ch +Ch c =1 for any event ; (iii) Ch 1 Ch 2 for any real number set 1 2, for more details, see Liu (213a). Besides, the subadditivity of chance measure is proved by Hou (214), that is, Ch i Ch i for a sequence of events 1, 2,... Definition 8 (Liu 213a) An uncertain random variable is a measurable function ξ from a chance space (Ɣ, L, M) (, A, Pr) to the set of real numbers, i.e.,ξ B is an event for any Borel set B. Theorem 3 (Liu 213a) Let ξ 1,ξ 2,...,ξ n be uncertain random variables on the chance space (Ɣ, L, M) (, A, Pr) and let f be a measurable function. Then ξ = f (ξ 1,ξ 2,...,ξ n ) is an uncertain random variable determined by for all (γ, ω) Ɣ. ξ(γ,ω) = f (ξ 1 (γ, ω), ξ 2 (γ, ω),..., ξ n (γ, ω)) To describe an uncertain random variable, Liu (213b) presented a definition of chance distribution.

21 H. Ahmadzade et al. Definition 9 (Liu 213b) Let ξ be an uncertain random variable. Then its chance distribution is defined by for any x R. (x) = Chξ x The chance distribution of a random variable is just its probability distribution, and the chance distribution of an uncertain variable is just its uncertainty distribution. Theorem 4 (Liu 213b) Let η 1,η 2,...,η m be independent random variables with probability distributions 1, 2,..., m, respectively, and τ 1,τ 2,...,τ n be uncertain variables. Then the uncertain random variable ξ = f (η 1,η 2,...,η m,τ 1,τ 2,..., τ n ) has a chance distribution (x) = F(x, y 1,...,y m )d 1 (y 1 )...d m (y m ) R m where F(x, y 1,...,y m ) is the uncertainty distribution of uncertain variable f (η 1, η 2,...,η m,τ 1,τ 2,...,τ n ) for any real numbers y 1, y 2,...,y m. Definition 1 (Liu 213b) Letξ be an uncertain random variable. Then its expected value is defined by + E[ξ] = Chξ rdr Chξ rdr provided that at least one of the two integrals is finite. Let denote the chance distribution of ξ. Liu (213b) gave a formula to calculate the expected value of an uncertain random variable with its chance distribution, that is, E[ξ] = + (1 (x))dx (x)dx. Theorem 5 (Liu 213a) Let η 1,η 2,...,η m be independent random variables with probability distributions 1, 2,..., m, respectively, and τ 1,τ 2,...,τ n be independent uncertain variables (not necessarily independent). Then the uncertain random variable ξ = f (η 1,...,η m,τ 1,...,τ n ) has an expected value E[ξ] = E[ f (y 1,...,y m,τ 1,...,τ n )]d 1...d m R m where E[ f (y 1,...,y m,τ 1,...,τ n )] is the expected value of the uncertain variable f (y 1,...,y m,τ 1,...,τ n ) for any real numbers y 1,...,y m.

On the convergence of uncertain random sequences 211 Theorem 6 (Liu 213a, Linearity of Expected Value Operator) Assume η 1 and η 2 are random variables (not necessarily independent), τ 1 and τ 2 are independent uncertain variables, and f 1 and f 2 are measurable functions. Then E[ f 1 (η 1,τ 1 ) + f 2 (η 2,τ 2 )]=E[ f 1 (η 1,τ 1 )]+E[ f 2 (η 2,τ 2 )]. In order to study the properties of uncertain random sequences, we introduce the following concepts of convergence: Definition 11 An uncertain random sequence ξ n, n 1 is said to be almost surely convergent (a.s.) to an uncertain random variable ξ if there exists an event with Ch =1 such that for every (γ, ω). lim ξ n(γ, ω) ξ(γ,ω) = Definition 12 An uncertain random sequence ξ n, n 1 is said to be convergent in measure to an uncertain random variable ξ if for any ɛ>. lim Ch(γ, ω) ξ n(γ, ω) ξ(γ,ω) ɛ = Definition 13 (Liu 213b)Let ξ be an uncertain random variable with a finite expected value E[ξ]. Then the variance of ξ is V [ξ] =E [ (ξ E[ξ]) 2]. Since (ξ E[ξ]) 2 is a nonnegative uncertain random variable, we also have V [ξ] = + Ch(ξ E[ξ]) 2 xdx. How do we obtain variance from chance distribution? Since the chance measure is a subadditivity measure, the variance of uncertain random variable cannot be derived simply by the chance distribution. In this case, Guo and Wang (214) suggested a stipulation as follows: Stipulation 1 (Guo and Wang 214) Letξ be an uncertain random variable with a chance distribution. Ifξ has a finite expected value E[ξ], then V [ξ] = + ( 1 ( E[ξ]+ x ) + ( E[ξ] x )) dx. Ahmadzade and Sheng (214) gave some formulas to calculate the variance of an uncertain random variable with the chance distribution and the inverse chance distribution.

212 H. Ahmadzade et al. Theorem 7 (Ahmadzade and Sheng 214) Let η 1,η 2,...,η m be independent random variables with probability distributions 1, 2,..., m, and τ 1,τ 2,...,τ n be independent uncertain variables with uncertainty distributions ϒ 1,ϒ 2,...,ϒ n, respectively. Suppose that ξ = f (η 1,η 2,...,η m,τ 1,τ 2,...,τ n ). Then E [ (ξ E[ξ]) k] = R m 1 (F 1 (α, y 1,...,y m ) E[ξ]) k dαd 1 (y 1 )...d m (y m ), where F 1 (α, y 1,...,y m ) is the inverse of uncertainty distribution of uncertain variable f (η 1,η 2,...,η m,τ 1,τ 2,...,τ n ) for any real numbers y 1, y 2,...,y m. 3 Moment inequalities for uncertain random variables In this section, we obtain an inequality for central moment of uncertain random variables. Theorem 8 Let η 1,η 2,...,η m be independent random variables with probability distributions 1, 2,..., m, and τ 1,τ 2,...,τ m be independent uncertain variables with uncertainty distributions ϒ 1,ϒ 2,...,ϒ m, respectively. Suppose that f 1, f 2,..., f m are increasing functions. If ξ 1 = f 1 (η 1,τ 1 ), ξ 2 = f 2 (η 2,τ 2 ),...,ξ m = f m (η m,τ m ), then where S m = m ξ i. E [(S m E[S m ]) k] m m k 1 E [(ξ i Eξ i ) k] Proof By invoking Theorem 7, we obtain E [ (S m E[S m ]) k] = R m 1 (F 1 S m (α, y 1,...,y m ) E[S m ]) k dαd 1 (y 1 )...d m (y m ). By using Theorem 2 and Theorem 6, we conclude that F 1 S m (α, y 1,...,y m ) = Fξ 1 1 (α, y 1,...,y m ) + +Fξ 1 m (α, y 1,...,y m ), E[S m ]=E[ξ 1 ]+E[ξ 2 ]+ +E[ξ m ]. As an application of Jensen s inequality, we have ( ) k 1 m x i 1 m m m xi k,

On the convergence of uncertain random sequences 213 and consequently, ( m ) k x i m k 1 m x k i. (1) Thus, relation (1) and Theorem 7 imply that E [(S m E[S m ]) k] ( 1 m k = [Fξ 1 i (α, y 1,...,y m ) E[ξ i ]) dαd 1 (y 1 )...d m (y m ) R m m k 1 m 1 m = m k 1 E R m (F 1 ξ i (α, y 1,...,y m ) E[ξ i ]) k dαd 1 (y 1 )...d m (y m ) [ (ξ i E[ξ i ]) k]. Theorem 9 Let η 1,η 2,...,η m be independent random variables with probability distributions 1, 2,..., m, and τ 1,τ 2,...,τ m be independent uncertain variables with uncertainty distributions ϒ 1,ϒ 2,...,ϒ m, respectively. Suppose that f 1, f 2,..., f m are increasing functions. If ξ 1 = f 1 (η 1,τ 1 ), ξ 2 = f 2 (η 2,τ 2 ),...,ξ m = f m (η m,τ m ), then where S m = m ξ i. Var[S m ] m m Var[ξ i ], Proof By setting k = 2 in Theorem 8, the proof is straightforward. Remark 1 In Theorem 9, if uncertain random variables degenerate into uncertain variables and m = 2, then the results in Yao (215) are obtained. 4 Convergence of uncertain random sequence In this section, we establish a version of Borel Cantelli lemma as Theorem 1. By using this theorem, we obtain some results about convergence of uncertain random variables. It is mentioned that throughout this section, under the assumption that the uncertain measure is continuous, several convergence theorems are derived. Furthermore, uncertain variables are defined on a continuous uncertainty space.

214 H. Ahmadzade et al. Proposition 1 Suppose that n, n 1 is a sequence of events such that for any ω, n (ω), n 1 is a sequence of independent uncertain events. Then Ch m=1 n=m n = lim m Ch n, n=m where n (ω) =γ Ɣ (γ, ω) n. Proof Since i (ω), n 1 is a non-increasing sequence of independent uncertain events and uncertain measure M is assumed continuous, we obtain M γ Ɣ (γ, ω) i = M γ Ɣ (γ, ω) i. n 1 n=1 Also, it is easy to see that ω M γ Ɣ (γ, ω) i x n 1 = ω M γ Ɣ (γ, ω) i x. n=1 Therefore, Pr ω M γ Ɣ (γ, ω) i x n=1 = Pr ω M γ Ɣ (γ, ω) i x n 1 = Pr ω M γ Ɣ (γ, ω) i x. n=1 Since ω M γ Ɣ (γ, ω) i x, n 1 is a non-increasing sequence, we get lim ω M γ Ɣ (γ, ω) i x n=1 = ω M γ Ɣ (γ, ω) i x.

On the convergence of uncertain random sequences 215 Furthermore, by using continuity property of probability measure, we obtain Pr lim ω M γ Ɣ (γ, ω) i x = lim Pr ω M γ Ɣ (γ, ω) i x. By definition of chance measure, we get Ch n i.o. =Ch i n=1 1 = Pr ω M γ Ɣ (γ, ω) i x dx n=1 1 = Pr ω M γ Ɣ (γ, ω) i x dx n 1 1 = Pr ω M γ Ɣ (γ, ω) i x dx n=1 1 = Pr ω lim M γ Ɣ (γ, ω) i x dx 1 = lim Pr ω M γ Ɣ (γ, ω) i x dx = lim Ch i Corollary 1 Suppose that τ n, n 1 is a sequence of independent uncertain variables and η n, n 1 is a sequence of random variables. Let ξ n, n 1 be a sequence of uncertain random variables such that ξ i = f i (η i,τ i ), i = 1, 2,...and ξ be an uncertain random variable (ξ may be constant).then Ch m=1 n=m ξ n ξ >ɛ = lim m Ch ξ n ξ >ɛ. n=m Proof Theorem 3 implies that ξ n ξ, n 1 is a sequence of uncertain random variables. Thus, by using Definition 9 and setting B = (ɛ, ), we conclude that n = ξ n ξ >ɛ is an event. Therefore, invoking Proposition 1 completes the proof.

216 H. Ahmadzade et al. Theorem 1 Suppose that n, n 1 is a sequence of events such that for any ω, n (ω), n 1 is a sequence of independent uncertain events, where n (ω) = γ Ɣ (γ, ω) n. If Ch n <, then Ch n i.o. =. n=1 Proof For each ω, it follows from the subadditivity of uncertain measure that M γ Ɣ (γ, ω) i M γ Ɣ (γ, ω) i. Thus, for any real number x, wehave ω M γ Ɣ (γ, ω) i x ω M γ Ɣ (γ, ω) i x. Monotonicity of probability measure implies that Pr ω M γ Ɣ (γ, ω) i x Pr ω M γ Ɣ (γ, ω) i x. By definition of chance measure and Proposition 1, we get Ch n i.o. =Ch = lim lim lim = lim n=1 1 1 1 i Pr ω M γ Ɣ (γ, ω) i x dx Pr ω M γ Ɣ (γ, ω) i x dx Pr ω M γ Ɣ (γ, ω) i x dx 1 Pr ω M γ Ɣ (γ, ω) i x dx

On the convergence of uncertain random sequences 217 = lim = lim 1 Ch i. Pr ω M γ Ɣ (γ, ω) i x dx Since n=1 Ch n <, we conclude that lim Ch i =. Thus, this completes the proof. Corollary 2 Suppose that τ n, n 1 is a sequence of independent uncertain variables and η n, n 1 is a sequence of random variables. Let ξ n, n 1 be a sequence of uncertain random variables such that ξ i = f i (η i,τ i ), i = 1, 2,...and ξ be an uncertain random variable (ξ may be constant)such that ξ = f (η, τ) and τ be independent of τ 1,τ 2,...,τ n.if Ch ξ n ξ >ɛ <, then n=1 Ch ξ n ξ >ɛ i.o. =. Proof Definition 9 and Theorem 3 imply that n = ξ n ξ >ɛ is an event. Since τ n, n 1 is a sequence of independent uncertain variables, we conclude that n (ω), n 1 is a sequence of independent uncertain events, where n (ω) =γ Ɣ ξ n (γ, ω) ξ(γ,ω) >ɛ. Thus, Theorem 1 completes the proof. Theorem 11 Let η n, n 1 be a sequence of random variables and τ n, n 1 be a sequence of independent uncertain variables. Suppose that ξ i = f i (η i,τ i ), i = 1, 2,... where f 1, f 2,... are measurable functions, furthermore ξ is an uncertain random variable such that ξ = f (η, τ) and τ is independent of τ 1,τ 2,...,τ n. ξ n, n 1 converges a.s. to ξ( may be constant) if and only if sup ξ n ξ, m 1 converges n m in measure to zero. Proof The sequence ξ n, n 1 converges a.s. to ξ if and only if for any ɛ>, we have Ch ξ n ξ >ɛ =. m=1 n=m Independence of uncertain variables and Theorem 1 imply that ξ n (ω)= f n (η n (ω), τ n ), n 1 is a sequence of independent uncertain variables. Thus, Corollary 1 concludes that

218 H. Ahmadzade et al. Ch m=1 n=m ξ n ξ >ɛ = lim m Ch Therefore, ξ n, n 1 converges a.s. to ξ if and only if converges in measure to zero. ξ n ξ >ɛ. n=m sup ξ n ξ, m 1 n m Theorem 12 Let η n, n 1 be a sequence of random variables and τ n, n 1 be a sequence independent uncertain variables, furthermore ξ be an uncertain random variable such that ξ = f (η, τ) and τ be independent of τ 1,τ 2,...,τ n. Consider ξ i = f i (η i,τ i ), i = 1, 2,... If ξ n, n 1 converges a.s. to ξ( may be constant), then ξ n, n 1 converges in measure to ξ. Proof It follows from Corollary 1 and monotonicity of chance measure that lim Ch ξ m ξ >ɛ lim Ch ξ i ξ >ɛ m m i=m = Ch ξ i ξ >ɛ i.o. =. Theorem 13 Let η n, n 1 be a sequence of random variables and τ n, n 1 be a sequence of independent uncertain variables, furthermore ξ be an uncertain random variable such that ξ = f (η, τ) and τ be independent of τ 1,τ 2,...,τ n. Consider ξ i = f i (η i,τ i ), i = 1, 2,... If ξ n, n 1 converges in measure to ξ( maybe constant), then there exists a nondecreasing subsequence n k, k 1 such that ξ nk, k 1 converges a.s. to ξ. Proof For any ɛ>, since 1 2 k <ɛwhenever k > log(ɛ 1 ), by assumption, there log 2 exists a non-decreasing subsequence, n k, k 1 such that Ch ξ nk ξ > 12 k < 1 2 k. Consequently, k=1 Ch ξ nk ξ > 12 k < k=1 1 2 k <. Thus, Corollary 2 implies that ξ nk, k 1 converges a.s. to ξ.

On the convergence of uncertain random sequences 219 Theorem 14 Let η n, n 1 be a sequence of independent and identically distributed random variables and τ n, n 1 be a sequence of independent and identically distributed uncertain variables, furthermore ξ be an uncertain random variable such that ξ = f (η, τ) and τ be independent of τ 1,τ 2,...,τ n. Consider If ξ i = f (η i,τ i ), i = 1, 2,... n Var[ξ i ]=o(n), then S n /n, n 1 converges in measure to e 1, where e 1 = E[ξ 1 ]. Proof Since η n, n 1 is a sequence of identically distributed random variables and also, τ n, n 1 is a sequence of identically distributed uncertain variables, Theorem 5 implies that E[ξ i ]=e 1, i = 1, 2,... By using Markov s inequality and Theorem 9, we get S n Ch n e 1 >ɛ Var[S n] n 2 ɛ 2 n Var[ξ i ] nɛ 2. Remark 2 Throughout the paper, we obtain several convergence theorems based on the continuity assumption of uncertain measure. Furthermore, it is mentioned that, if continuity assumption is removed, then the results are not satisfied. Also, if uncertain random variables degenerate into uncertain variables, the results in Chen et al. (214) are obtained. 5 Conclusions This paper studied some properties of uncertain random sequences. We first proved an inequality for central moment of uncertain random variables. Based on the continuity assumption of uncertain measure, a version of the Borel Cantelli lemma was proved. Finally, several convergence theorems for uncertain random variables were established. It is mentioned that, if continuity assumption is removed, then the results are not satisfied. Furthermore, if uncertain random variables degenerate into uncertain variables, the results are hold. Acknowledgements This work was supported by National Natural Science Foundation of China Grant Nos. 6146286, 615635 and in part by Xinjiang University (No. BS1526).

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