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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 3, JUNE 2016 615 Law of Large Numbers for Uncertain Random Variables Kai Yao and Jinwu Gao Abstract—The law of large numbers in probability theory states that the average of random variables converges to its expected value in some sense under some conditions. Sometimes, random factors and human uncertainty exist simultaneously in complex systems, and a concept of uncertain random variable has been proposed to study this type of complex systems. This paper aims to provide a law of large numbers for uncertain random variables, which states that the average of uncertain random variables converges in distribution to an uncertain variable. As a byproduct, the convergence of a sequence of uncertain variables is also studied. Index Terms—Chance theory, law of large numbers, uncertain random variable, uncertainty theory. I. INTRODUCTION N probability theory, the law of large numbers ensures that the average of some random variables converges to a real number in some sense under some conditions. First pointed out by Cardano in the 16th century, the law of large numbers has been studied by a number of famous researchers, including Bernoulli, Poisson, Chebyshev, Markov, Borel, Cantelli, Kolmogorov, and Khinchin. Nowadays, many researchers stay focused on this problem. Baum and Katz [1] studied the convergence rate of the law of large numbers, and Hoffmann-Jørgensen and Pisier [2] studied the law of large numbers in a Banach space. Acosta [3] gave some inequalities about the law of large numbers for random vectors, and Uhlig [4] provided an application of law of large numbers in economics. For recent developments in the law of large numbers, see [5]–[7]. In order to model the fuzzy phenomena in daily life, Zadeh [8] proposed a concept of fuzzy set, and Zadeh [9] further founded a possibility theory. Fuzzy measures could be distinguished into many classes. Valášková and Struk [10], [11] classified the fuzzy measures into submeasure, supermeasure, submodular, supermodular, belief, plausibility, possibility, and necessity, and they showed that these classes are closed under the operations of distortion functions. Besides, both the Choquet integral and the Sugeno integral were further studied in [12] and [13]. The law of large numbers for fuzzy sets was first proposed by Fullér [14], and it was further generalized by Hong and coauthors [15], [16]. In order to model complex phenomena with both randomness I Manuscript received January 6, 2015; revised May 13, 2015 and June 24, 2015; accepted July 30, 2015. Date of publication August 7, 2015; date of current version May 30, 2016. This work was supported in part by the National Natural Science Foundation of China under Grant 61403360 and Grant 61374082. K. Yao is with the School of Economics and Management, University of Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]). J. Gao is with the School of Information, Renmin University of China, Beijing 100872, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2015.2466080 and fuzziness, Kwakernaak [17], [18] proposed a concept of fuzzy random variables. Then, Kruse [19] and Miyakoshi and Shimbo [20] proposed some laws of large numbers for fuzzy random variables. These results were further developed by Inoue [21], Kim [22], Joo and Kim [23], and so on. Except for randomness and fuzziness, human uncertainty is another source of indeterminacy. In order to deal with human’s belief degree that some events occur, an uncertainty theory was founded by Liu [24] in 2007, and refined by Liu [25] in 2009 based on normality, duality, subadditivity, and product axioms. Similar to probability theory, a concept of uncertain measure is to indicate the belief degree of possible events, and a concept of uncertain variable is to model the quantities under uncertain status in the framework of uncertainty theory. Many researchers have contributed a lot in this area over the past years. For example, Peng and Iwamura [26] gave a sufficient and necessary condition for a real function being an uncertainty distribution of an uncertain variable. Liu and Ha [27] proposed a formula to calculate the expected value of a function of some uncertain variables. Chen and Dai [28] verified that a normal uncertain variable possesses a maximum entropy given the expected value and variance. In a complex system, we may have a large samples on some components but have no samples on some other components. For the first class of components, we can obtain the probability distributions of their lifetimes via statistics, but for the second class, we can only get experts’ belief degree. Therefore, the system behaves both randomly and uncertainly and cannot be dealt with simply by probability theory or uncertainty theory. In order to model such systems, Liu [29], [30] founded a chance theory based on probability theory and uncertainty theory in 2013. A concept of chance space was proposed by Liu [29] as a product of probability space and uncertainty space, and a chance measure, as a generalization of probability measure and uncertain measure, was also defined to indicate the possibility that an uncertain random event happens. Then, a concept of uncertain random variable was proposed to model the quantities under uncertain and random condition, and chance distribution, expected value, and variance were employed to describe an uncertain random variable afterwards. In 2013, Liu [30] proposed uncertain random programming as a spectrum of mathematical programming involving uncertain random variables. Then, uncertain random multiobjective programming, uncertain random goal programming, and uncertain random multilevel programming were developed by Zhou et al. [31], Qin [32], and Ke et al. [33], [34], respectively. In addition, chance theory was also applied to risk analysis (see [35]), reliability analysis (see [36]), graph and network (see 1063-6706 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. 616 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 3, JUNE 2016 [37]), propositional logic (see [38]), as well as uncertain random process (see [39] and [40]). This paper will provide a law of large numbers for uncertain random variables. The rest of this paper is organized as follows. Section II will review some basic concepts about uncertain variable and uncertain random variable. Then, the concept of convergence in distribution for uncertain random variables is investigated in Section III. After that, the integral of a time function of an uncertain variable is studied in Section IV, and its uncertainty distribution and expected value are studied. Then, the law of large numbers for uncertain random variables is proved in Section V, which shows the average of uncertain random variables converges in distribution to an uncertain variable. Finally, some remarks are made in Section VI. II. PRELIMINARY This section will introduce some basic definitions and theorems about uncertain variables and uncertain random variables. An uncertainty distribution Φ is said to be regular if its inverse function Φ−1 (α) exists and is unique for each α ∈ (0, 1). An uncertain variable with a regular uncertainty distribution is usually called a regular uncertain variable in abbreviation. Inverse uncertainty distribution plays an important role in the operation of independent uncertain variables. Definition 5 (see[25]): The uncertain variables ξ1 , ξ2 , . . . , ξm on the same uncertainty space are said to be independent if m m M (ξi ∈ Bi ) = M{ξi ∈ Bi } i=1 i=1 for any Borel sets B1 , B2 , . . . , Bm of real numbers. Theorem 1 (see [41]): Let ξ1 , ξ2 , . . . , ξn be independent uncertain variables with regular uncertainty distributions Φ1 , Φ2 , . . . , Φn , respectively. If the function f (x1 , x2 , . . . , xn ) is strictly increasing with respect to x1 , x2 , . . . , xm and strictly decreasing with respect to xm +1 , xm +2 , . . . , xn , then ξ = f (ξ1 , ξ2 , . . . , ξn ) A. Uncertain Variable Definition 1 (see[24]): Let L be a σ-algebra on a nonempty set Γ. A set function M : L → [0, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1 (normality axiom): M{Γ} = 1 for the universal set Γ. Axiom 2 (duality axiom): M{Λ} + M{Λc } = 1 for any event Λ. Axiom 3 (subadditivity axiom): For every countable sequence of events Λ1 , Λ2 , . . . , we have ∞ ∞ M Λi ≤ M {Λi } . i=1 i=1 Then, the triple (Γ, L, M) is called an uncertainty space. Definition 2 (see [25]): Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, . . .. Then, the measure M on the product σ-algebra is called the product uncertain measure if it satisfies ∞ ∞ M Λk = Mk {Λk } k =1 k =1 where Λk are arbitrarily chosen events from Lk for k = 1, 2, . . ., respectively. Definition 3 (see [24]): An uncertain variable ξ is a measurable function from Γ to the set of real numbers , i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ ∈ Γ | ξ(γ) ∈ B} is an event. In order to describe an uncertain variable, a concept of uncertainty distribution is defined as follows. Definition 4 (see [24]): Let ξ be an uncertain variable. Then, its uncertainty distribution is defined by Φ(x) = M{ξ ≤ x} for any real number x. is an uncertain variable with inverse uncertainty distribution −1 Φ−1 (α) = f Φ−1 1 (α), . . . , Φm (α), −1 Φ−1 m +1 (1 − α), . . . , Φn (1 − α) . Definition 6 (see [24]): Let ξ be an uncertain variable. Then, the expected value of ξ is defined by 0 +∞ M{ξ ≥ x}dx − M{ξ ≤ x}dx E[ξ] = −∞ 0 provided that at least one of the two integrals is finite. In order to calculate the expected value of a function of an uncertain variable, Liu and Ha [27] provided the following theorem. Theorem 2 (see [27]): Let ξ be an uncertain variable with an uncertainty distribution Φ. If f (x) is a monotone function such that E[f (ξ)] exists, then +∞ f (x)dΦ(x). E[f (ξ)] = −∞ B. Uncertain Random Variable Let (Γ, L, M) be an uncertainty space, and (Ω, A, Pr) be a probability space. Then (Γ, L, M) × (Ω, A, Pr) = (Γ × Ω, L × A, M × Pr) is called a chance space. Definition 7 (see [29]): Let (Γ, L, M) × (Ω, A, Pr) be a chance space, and Θ ∈ L × A be an uncertain random event. Then, the chance measure Ch of Θ is defined by 1 Ch{Θ} = Pr{ω ∈ Ω | M{γ ∈ Γ | (γ, ω) ∈ Θ} ≥ r}dr. 0 Liu [29] verified that the chance measure Ch satisfies normality, duality, and monotonicity properties, that is, 1) Ch{Γ × Ω} = 1; 2) Ch{Θ} + Ch{Θc } = 1 for any event Θ; and 3) Ch{Θ1 } ≤ Ch{Θ2 } for any events Θ1 and Θ2 with YAO AND GAO: LAW OF LARGE NUMBERS FOR UNCERTAIN RANDOM VARIABLES Θ1 ⊂ Θ2 . Besides, Hou [42] proved the subadditivity of chance measure, that is, ∞ ∞ Θi ≤ Ch{Θi } Ch i=1 i=1 for a sequence of uncertain random events Θ1 , Θ2 , . . .. Definition 8 (see [29]): An uncertain random variable ξ is a measurable function from a chance space (Γ, L, M) × (Ω, 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. Random variable and uncertain variable can be regarded as special cases of uncertain random variable. Let η be a random variable, τ be an uncertain variable, and f be a measurable function. Then, f (η, τ ) is an uncertain random variable. Definition 9 (see [29]): Let ξ be an uncertain random variable. Then, its chance distribution is defined by Φ(x) = Ch {ξ ≤ x} ∀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. For a random variable η with a probability distribution Ψ, an uncertain variable τ , and a measurable function f , Liu [30] proved that f (η, τ ) has a chance distribution +∞ F (x, y)dΨ(x) Φ(y) = −∞ where F (x, y) = M{f (x, τ ) ≤ y} is the uncertainty distribution of f (x, τ ). Definition 10 (see[29]): Let ξ be an uncertain random variable. Then, its expected value is defined by 0 +∞ Ch{ξ ≥ r}dr − Ch{ξ ≤ r}dr E[ξ] = −∞ 0 provided that at least one of the two integrals is finite. If the uncertain random variable ξ has a chance distribution Φ, then +∞ 0 E[ξ] = (1 − Φ(r))dr − Φ(r)dr. −∞ 0 For a random variable η and an uncertain variable τ , Liu [30] proved that E[η + τ ] = E[η] + E[τ ] and E[η × τ ] = E[η] × E[τ ]. III. CONVERGENCE IN DISTRIBUTION This section introduces a concept of convergence in distribution for a sequence of uncertain random variables and illustrates the concept via some examples. Definition 11: Let ξ, ξ1 , ξ2 , . . . be a sequence of uncertain random variables with chance distributions Φ, Φ1 , Φ2 , . . ., respectively. Then, {ξi } is said to converge in distribution to ξ if lim Φi (x) = Φ(x) i→∞ 617 for every x ∈ at which Φ(x) is continuous. Remark 1: The concept of convergence in distribution for uncertain random variables coincides with the concepts of convergence in distribution for random variables and for uncertain variables, respectively. If {ηi } is a sequence of random variables converging in distribution to η, then {ηi } is also a sequence of uncertain random variables that converges in distribution to η. If {τi } is a sequence of uncertain variables converging in distribution to τ , then {τi } is also a sequence of uncertain random variables that converges in distribution to τ. Example 1: Let ξi be an uncertain random variable with a chance distribution ⎧ 0, if x < 1 − 1/i ⎪ ⎪ ⎪ ⎨ ix − i + 1 Φi (x) = , if 1 − 1/i ≤ x ≤ 2 + 1/i ⎪ i+2 ⎪ ⎪ ⎩ 1, if x > 2 + 1/i for i = 1, 2, . . ., and let ξ be an uncertain random variable with a chance distribution ⎧ 0, if x < 1 ⎪ ⎪ ⎨ Φ(x) = x − 1, if 1 ≤ x ≤ 2 ⎪ ⎪ ⎩ 1, if x > 2. Since Φi (x) → Φ(x) for every real number x as i → ∞, the uncertain random sequence {ξi } converges in distribution to ξ. Lemma 1: Let {τi } be a sequence of uncertain variables converging in distribution to a regular uncertain variable τ . If f is a continuous and strictly monotone function, then the uncertain sequence {f (τi )} converges in distribution to f (τ ). Proof: The proof breaks into two cases according to the monotonicity of f . Case 1: Assume f (x) is a strictly increasing function with respect to x. For simplicity, we define an inverse function ⎧ −∞, if z ≤ inf{f (x)|x ∈ } ⎪ ⎪ ⎨ −1 if z = f (y) f (z) = y, ⎪ ⎪ ⎩ +∞, if z ≥ sup{f (x)|x ∈ }. Let Φ denote the uncertainty distribution of τ , and Φi denote the uncertainty distributions of τi for i = 1, 2, . . .. Then, we have M{f (τ ) ≤ z} = M{τ ≤ f −1 (z)} = Φ f −1 (z) M{f (τi ) ≤ z} = M{τi ≤ f −1 (z)} = Φi f −1 (z) . Since the function Φ isa regular uncertainty distribution, we have Φi f −1 (z) → Φ f −1 (z) as i → ∞ for each real number z. Hence, the sequence {f (τi )} converges in distribution to f (τ ). Case 2: Assume that f (x) is a strictly decreasing function with respect to x. Then, −f (x) is a strictly increasing function, and the sequence {−f (τi )} converges in distribution to −f (τ ) as i → ∞ according to Case 1. As a result, the sequence {f (τi )} converges in distribution to f (τ ) as i → ∞. The proof is complete. Theorem 3: Let {ηi } be a sequence of random variables converging in distribution to η, and {τi } be a sequence of uncertain 618 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 3, JUNE 2016 variables converging in distribution to a regular uncertain variable τ . Assume that f (x, y) is a continuous function, and it is strictly monotone with respect to y for every real number x. Then, the uncertain random variables f (ηi , τi ) → f (η, τ ) as i → ∞ in the sense of convergence in distribution. Proof: Let F (x, y) = M{f (x, τ ) ≤ y} denote the uncertainty distribution of f (x, τ ), and Fi (x, y) = M{f (x, τi ) ≤ y} denote the uncertainty distributions of f (x, τi ) for i = 1, 2, . . .. Then, according to Lemma 1, for any real numbers x and y, we have Fi (x, y) → F (x, y) as i → ∞. Let Ψ denote the probability distribution of η, and Ψi denote the probability distributions of ηi for i = 1, 2, . . .. Then, according to the definition of convergence in distribution for random variables, we have Ψi (x) → Ψ(x) as i → ∞ for every x at which Ψ(x) is continuous. Note that f (η, τ ) has a chance distribution +∞ F (x, y)dΨ(x) = E[F (η, y)] Ch{f (η, τ ) ≤ y} = for any given real number y. Proof: Since the function f (x, y) is strictly increasing with respect to y for every real number x, the function +∞ f (x, y)dΦ(x) −∞ is also increasing with respect to y. Then, on the one hand, for any γ ∈ Γ such that τ (γ) ≤ y, we have +∞ +∞ f (x, τ (γ))dΦ(x) ≤ f (x, y)dΦ(x). −∞ −∞ As a result, we get +∞ f (x, τ )dΦ(x) ≤ −∞ +∞ −∞ which implies +∞ M f (x, τ )dΦ(x) ≤ −∞ −∞ E[Fi (ηi , y)] → E[F (η, y)] as i → ∞ for each y, the uncertain random sequence {f (ηi , τi )} converges in distribution to f (η, τ ). Example 2: Let {ηi } be a sequence of random variables converging in distribution to η, and {τi } be a sequence of uncertain variables converging in distribution to τ . Then, the uncertain random variables ηi + τi → η + τ as i → ∞ in the sense of convergence in distribution. Example 3: Let {ηi } be a sequence of positive random variables converging in distribution to η, and {τi } be a sequence of uncertain variables converging in distribution to τ . Then, the uncertain random variables ηi τi → ητ as i → ∞ in the sense of convergence in distribution. Example 4: Let {ηi } be a sequence of random variables converging in distribution to η, and {τi } be a sequence of uncertain variables converging in distribution to τ . Then, the uncertain random variables ηi ∨ τi → η ∨ τ as i → ∞ in the sense of convergence in distribution, and the uncertain random variables ηi ∧ τi → η ∧ τ as i → ∞ in the sense of convergence in distribution. IV. INTEGRAL OF UNCERTAIN VARIABLE In this section, we introduce the integral of an uncertain variable and obtain its uncertainty distribution and expected value. Theorem 4: Let τ be an uncertain variable with an uncertainty distribution Ψ, and Φ be a probability distribution of some random variables. If f (x, y) is a strictly increasing function with respect to y for every real number x, then +∞ +∞ f (x, τ )dΦ(x) ≤ f (x, y)dΦ(x) = Ψ(y) M −∞ −∞ f (x, y)dΦ(x) ≥ M{τ ≤ y} = Ψ(y). (1) On the other hand, for any γ ∈ Γ such that τ (γ) > y, we have +∞ +∞ f (x, τ (γ))dΦ(x) > f (x, y)dΦ(x). −∞ for i = 1, 2, . . .. Since +∞ −∞ −∞ and f (ηi , τi ) have chance distributions +∞ Fi (x, y)dΨi (x) = E[Fi (ηi , y)] Ch{f (ηi , τi ) ≤ y} = f (x, y)dΦ(x) ⊃ {τ ≤ y} −∞ As a result, we get +∞ f (x, τ )dΦ(x) > −∞ +∞ −∞ f (x, y)dΦ(x) ⊃ {τ > y} which implies +∞ f (x, τ )dΦ(x) > M −∞ +∞ f (x, y)dΦ(x) −∞ ≥ M{τ > y} = 1 − Ψ(y). Then M +∞ −∞ f (x, τ )dΦ(x) ≤ +∞ f (x, y)dΦ(x) −∞ ≤ M{τ ≤ y} = Ψ(y) (2) by the duality of uncertain measure. Combining inequalities (1) and (2), we have +∞ +∞ M f (x, τ )dΦ(x) ≤ f (x, y)dΦ(x) = Ψ(y). −∞ −∞ Theorem 5: Let τ be an uncertain variable with an uncertainty distribution Ψ, and Φ be a probability distribution of some random variables. If f (x, y) is a strictly monotone function with respect to y for every real number x, then +∞ +∞ +∞ f (x, τ )dΦ(x) = f (x, y)dΦ(x)dΨ(y). E −∞ −∞ −∞ Proof: Since the function f (x, y) is strictly monotone with respect to y for every real number x, the function +∞ f (x, y)dΦ(x) −∞ YAO AND GAO: LAW OF LARGE NUMBERS FOR UNCERTAIN RANDOM VARIABLES is also monotone with respect to y. According to Theorem 2, we have +∞ +∞ +∞ E f (x, τ )dΦ(x) = f (x, y)dΦ(x)dΨ(y). −∞ −∞ −∞ The proof is complete. V. LAW OF LARGE NUMBERS In this section, we prove the law of large numbers, which shows the average of uncertain random variables converges in distribution to an uncertain variable under some conditions. Theorem 6 (Law of Large Numbers): Let η1 , η2 , . . . be a sequence of iid random variables with a common probability distribution Φ, and τ1 , τ2 , . . . be a sequence of iid regular uncertain variables. Define Sn = f (η1 , τ1 ) + f (η2 , τ2 ) + · · · + f (ηn , τn ) for n ≥ 1, where f (x, y) is a continuous and strictly monotone function. Then +∞ Sn → f (x, τ1 )dΦ(x) n −∞ as n → ∞ in the sense of convergence in distribution. Proof: The proof breaks into two cases according to the monotonicity of the function f . Case 1: Assume that f (x, y) is strictly increasing with respect to y for every x ∈ . Since f (x, y) is also a continuous function, the function +∞ f (x, y)dΦ(x) F (y) = −∞ is a continuous and strictly verse function ⎧ −∞, ⎪ ⎪ ⎨ −1 F (z) = y, ⎪ ⎪ ⎩ +∞, increasing function. Define an inif z ≤ inf{F (y)|y ∈ } if F (y) = z if z ≥ sup{F (y)|y ∈ }. Then, we have +∞ M f (x, τ1 )dΦ(x) ≤ z −∞ =M f (x, τ1 )dΦ(x) ≤ +∞ −∞ = Ψ F −1 (z) +∞ f x, F −1 (z) dΦ(x) −∞ (3) 619 that f η1 , F −1 (z − ε) + · · · + f ηn , F −1 (z − ε) ≤z Pr n f η1 , F −1 (z − ε) + · · · + f ηn , F −1 (z − ε) = Pr n +∞ −1 ≤ f x, F (z − ε) dΦ(x) + ε −∞ ≥1−ε for any n ≥ N1 . Furthermore, we have Sn ≤z n 1 f (η1 , τ1 ) + · · · + f (ηn , τn ) = ≤ z ≥ r dr Pr M n 0 1 f (η1 , F −1 (z − ε)) + · · · + f (ηn , F −1 (z − ε)) ≤z ≥ Pr n 0 f (η1 , τ1 ) + · · · + f (ηn , τn ) ≤ z ≥ r dr ∩ M n 1 f (η1 , F −1 (z − ε)) + · · · + f (ηn , F −1 (z − ε)) ≤z ≥ Pr n 0 f (η1 , τ1 ) + · · · + f (ηn , τn ) ≤ ∩ M n f (η1 ,F −1 (z − ε))+· · ·+f (ηn , F −1 (z − ε)) ≥ r dr n 1 f (η1 , F −1 (z − ε)) + · · · + f (ηn , F −1 (z − ε)) ≤z Pr = n 0 ∩ Ψ F −1 (z − ε) ≥ r dr Ψ (F −1 (z −ε ) ) f (η1 , F −1 (z − ε)) + ··· = Pr n 0 f (ηn , F −1 (z − ε)) + ≤ z dr n −1 ≥ Ψ F (z − ε) (1 − ε) Ch for any n ≥ N1 . Thus, we obtain that Sn ≤ z ≥ Ψ F −1 (z) lim Ch n →∞ n (4) by Theorem 4 where Ψ is an uncertainty distribution of τi . For z and any number given ε > 0, since any given real f η1 , F −1 (z − ε) , f η2 , F −1 (z − ε) , . . . are a sequence of iid random variables, we have f η1 , F −1 (z − ε) + · · · + f ηn , F −1 (z − ε) n +∞ → f x, F −1 (z − ε) dΦ(x), a.s. for any z ∈ . On the other hand, for any given real number z and any given ε > 0, since f η1 , F −1 (z + ε) , f η2 , F −1 (z + ε) , . . . are a sequence of iid random variables, we have f η1 , F −1 (z + ε) + · · · + f ηn , F −1 (z + ε) n +∞ −1 → f x, F (z + ε) dΦ(x), a.s as n → ∞ by the strong law of large numbers for random variables. In other words, there exists a positive number N1 such as n → ∞ by the strong law of large numbers for random variables. In other words, there exists a positive number N2 such −∞ −∞ 620 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 3, JUNE 2016 that f η1 , F −1 (z + ε) + · · · + f ηn , F −1 (z + ε) Pr >z n f η1 , F −1 (z + ε) + · · · + f ηn , F −1 (z + ε) ≥ Pr n +∞ > f x, F −1 (z + ε) dΦ(x) − ε −∞ ≥1−ε for any n ≥ N2 . Furthermore, we have Sn >z n 1 f (η1 , τ1 ) + · · · + f (ηn , τn ) = > z ≥ r dr Pr M n 0 1 −1 f (η1 , F (z + ε)) + · · · + f (ηn , F −1 (z + ε)) >z ≥ Pr n 0 f (η1 , τ1 ) + · · · + f (ηn , τn ) > z ≥ r dr ∩ M n 1 f (η1 , F −1 (z + ε)) + · · · + f (ηn , F −1 (z + ε)) >z ≥ Pr n 0 f (η1 , τ1 ) + · · · + f (ηn , τn ) > ∩ M n f (η1 , F −1 (z + ε))+· · ·+f (ηn , F −1 (z + ε)) ≥ r dr n 1 f (η1 , F −1 (z + ε)) + · · · + f (ηn , F −1 (z + ε)) >z = Pr n 0 ∩ 1 − Ψ F −1 (z + ε) ≥ r dr 1 −Ψ (F −1 (z + ε ) ) f (η1 , F −1 (z + ε)) + ··· = Pr n 0 f (ηn , F −1 (z + ε)) + > z dr n −1 ≥ 1 − Ψ F (z + ε) (1 − ε) Ch for any n ≥ N2 . By the duality of chance measure, we have Sn ≤ z ≤ 1 − 1 − Ψ F −1 (z + ε) (1 − ε) Ch n for any n ≥ N2 . Thus, we obtain that Sn ≤ z ≤ Ψ F −1 (z) lim Ch n →∞ n +∞ −∞ for every point z ∈ , which is apparently equivalent to +∞ Sn >z =M f (x, τ1 )dΦ(x) > z . lim Ch n →∞ n −∞ By the duality of uncertain measure and chance measure, we further have +∞ Sn ≤z =M lim Ch f (x, τ1 )dΦ(x) ≤ z . n →∞ n −∞ The proof is thus complete. Remark 2: It is believed that Theorem 6 holds in a general situation: τi is not necessarily regular, and f is not necessarily continuous. However, a rigorous proof has not been given for this general case so far. Example 5: For a sequence of iid random variables η1 , η2 , . . ., write Sn = η1 + η2 + · · · + ηn for n ≥ 1. Then, it follows from Theorem 6 that Sn /n converges in distribution to E[η1 ]. Example 6: For a sequence of iid uncertain variables τ1 , τ2 , . . ., write Sn = τ1 + τ2 + · · · + τn for n ≥ 1. Then, it follows from Theorem 6 that Sn /n converges in distribution to τ1 . Example 7: Let η1 , η2 , . . . be a sequence of iid random variables, and τ1 , τ2 , . . . be a sequence of iid uncertain variables. Define Sn = (η1 + τ1 ) + · · · + (ηn + τn ) for n ≥ 1. Then Sn → E[ξ1 ] + τ1 n in the sense of convergence in distribution as n → ∞. Example 8: Let η1 , η2 , . . . be a sequence of iid positive random variables, and τ1 , τ2 , . . . be a sequence of iid uncertain variables. Define Sn = η1 τ1 + · · · + ηn τn for n ≥ 1. Then Sn → E[ξ1 ]τ1 n in the sense of convergence in distribution as n → ∞. VI. CONCLUSION (5) for any z ∈ . By inequalities (4) and (5), we have Sn ≤ z = Ψ F −1 (z) lim Ch n →∞ n =M is a strictly increasing function with respect to y. By a similar proof to Case 1, we have +∞ Sn < −z = M − f (x, τ1 )dΦ(x) < −z lim Ch − n →∞ n −∞ f (x, τ1 )dΦ(x) ≤ z and Case 1 is thus proved. Case 2: Assume that f (x, y) is strictly decreasing with respect to y for every x ∈ . 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Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin, Germany: Springer-Verlag, 2010. [42] Y. C. Hou, “Subadditivity of chance measure,” J. Uncertain. Anal. Appl., vol. 2, p. 14, Jun. 2014. Kai Yao received the B.S. degree from Nankai University, Tianjin, China, in 2009, and the Ph.D. degree from Tsinghua University, Beijing, China, in 2013. He is currently an Assistant Professor with the School of Management, University of Chinese Academy of Sciences, Beijing, China. He has authored or coauthored more than 30 articles on several journals including IEEE TRANSACTIONS ON FUZZY SYSTEMS, Knowledge-Based Systems, Applied Soft Computing, Applied Mathematical Modelling, Applied Mathematics and Computation, Fuzzy Optimization and Decision Making, and Soft Computing. His current research interests include uncertain renewal processes, uncertain systems, uncertain differential equations, and their applications. Jinwu Gao received the B.S. degree in mathematics from Shaanxi Normal University, Xi’an, China, in 1996, and the M.S. and Ph.D. degrees in mathematics from Tsinghua University, Beijing, China, in 2005. Dr. Gao is currently an Associate Professor with the School of Information, Renmin University of China. His current research interests include fuzzy systems, uncertain systems, and their application in optimization, game theory, and finance. He has authored or coauthored more than 30 papers that have appeared in IEEE TRANSACTIONS ON FUZZY SYSTEMS, Soft Computing, Fuzzy Optimization and Decision Making, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Iranian Journal of Fuzzy Systems, Computer & Mathematics with Applications, and other publications. Dr. Gao has been the Co-Editor-in-Chief of the Journal of Uncertain Systems since 2011 and the Executive-Editor-in-Chief of the Journal of Uncertainty Analysis and Applications since 2013. He has served as the Vice President and President of Intelligent Computing Chapter of the Operations Society of China since 2007 and 2015, respectively, the President of International Consortium for Electronic Business from 2012 to 2013, the Vice President of International Consortium for Uncertainty Theory since 2013, and the Vice President of International Association for Information and Management Science from 2010 to 2013.