Limit Theorems for Negatively Dependent Fuzzy Set-Valued Random Variables

We all know that strong laws of large numbers are one of the most important theories in probability. For independent set-valued random variables, many limit results have been obtained (cf. (Artstein & Vitale, 1975), (Cressie, 1978), (Hiai, 1984), (Taylor & Inoue, 1985), (Puri & Ralescu, 1983)). About the convergence theorems of fuzzy set-valued random variables, Klement et al proved the strong laws of large numbers (SLLN) for independent identically distributed (i.i.d) fuzzy set-valued random variables in the sense of d1 H . Colubi et al. obtained the SLLN for i.i.d fuzzy set-valued random variables with respect to d∞ H in (Colubi, López-Dı́az, Dominguez-Menchero & Gil, 1999), where the underlying space is Rd. Li and Ogura proved the same SLLN as in (Colubi, López-Dı́az, Dominguez-Menchero & Gil, 1999) by using a new embedding method (Li, Ogura & Kreinovich, 2002, Theorem 6.2.6). In 1991, Inoue (Inoue, 1991) extended the SLLN of set-valued case given by Taylor and Inoue (Taylor & Inoue, 1985) to the case of only independent fuzzy set-valued random variables in the sense of d1 H . Li and Ogura (Li & Ogura, 2003) proved the SLLN of (Inoue, 1991) in the sense of d∞ H . In practice, the random variables are not always independent. So it is necessary to discuss the limit theorems for dependent random variables. Bozorgnia, Patterson and Taylor discussed the properties for negatively dependent random variables in (Bozorgnia, Patterson, & Taylor, 1993), and proved the laws of large number for negative dependence random variables in (Bozorgnia, Patterson & Taylor, 1992), where the random variables are single-valued. In (Guan, 2014), Guan and Sun discussed the property of set-valued random variables in real space R, and proved the weak convergence theorem for dependent set-vallued random variables in the sense of Hausdorff metric. In (Guan, 2016), Guan and Wan proved the strong law of large number for set-valued dependent random variables in the sense of Hausdorffmetric. In 2016, Shen and Guan (Shen & Guan, 2016) proved the strong laws of large numbers for independent fuzzy set-valued random variables, where the underlying space is Gα space. In this paper, we are concerned with the weak and strong limit theorems for dependent fuzzy set-valued random variables in the sense of d∞ H . This paper is organized as follows. In section 2, we shall briefly introduce some definitions and basic results of set-valued random variables and fuzzy set-valued random variables. In section 3, we shall give basic definition and discuss the properties of fuzzy set-valued negatively dependent random variables. And at last, we shall prove the weak and strong laws of large numbers for weighted sums of fuzzy set-valued negatively dependent random variables.


Introduction
We all know that strong laws of large numbers are one of the most important theories in probability.For independent set-valued random variables, many limit results have been obtained (cf.(Artstein & Vitale, 1975), (Cressie, 1978), (Hiai, 1984), (Taylor & Inoue, 1985), (Puri & Ralescu, 1983)).About the convergence theorems of fuzzy set-valued random variables, Klement et al proved the strong laws of large numbers (SLLN) for independent identically distributed (i.i.d) fuzzy set-valued random variables in the sense of d 1 H . Colubi et al. obtained the SLLN for i.i.d fuzzy set-valued random variables with respect to d ∞ H in (Colubi, López-Díaz, Dominguez-Menchero & Gil, 1999), where the underlying space is R d .Li and Ogura proved the same SLLN as in (Colubi, López-Díaz, Dominguez-Menchero & Gil, 1999) by using a new embedding method (Li, Ogura & Kreinovich, 2002, Theorem 6.2.6).In 1991, Inoue (Inoue, 1991) extended the SLLN of set-valued case given by Taylor and Inoue (Taylor & Inoue, 1985) to the case of only independent fuzzy set-valued random variables in the sense of d 1 H . Li and Ogura (Li & Ogura, 2003) proved the SLLN of (Inoue, 1991) in the sense of d ∞ H .In practice, the random variables are not always independent.So it is necessary to discuss the limit theorems for dependent random variables.Bozorgnia, Patterson and Taylor discussed the properties for negatively dependent random variables in (Bozorgnia, Patterson, & Taylor, 1993), and proved the laws of large number for negative dependence random variables in (Bozorgnia, Patterson & Taylor, 1992), where the random variables are single-valued.In (Guan, 2014), Guan and Sun discussed the property of set-valued random variables in real space R, and proved the weak convergence theorem for dependent set-vallued random variables in the sense of Hausdorff metric.In (Guan, 2016), Guan and Wan proved the strong law of large number for set-valued dependent random variables in the sense of Hausdorff metric.In 2016, Shen and Guan (Shen & Guan, 2016) proved the strong laws of large numbers for independent fuzzy set-valued random variables, where the underlying space is G α space.In this paper, we are concerned with the weak and strong limit theorems for dependent fuzzy set-valued random variables in the sense of d ∞ H .This paper is organized as follows.In section 2, we shall briefly introduce some definitions and basic results of set-valued random variables and fuzzy set-valued random variables.In section 3, we shall give basic definition and discuss the properties of fuzzy set-valued negatively dependent random variables.And at last, we shall prove the weak and strong laws of large numbers for weighted sums of fuzzy set-valued negatively dependent random variables.

Preliminaries on Fuzzy Set-Valued Random Variables
Throughout this paper, we assume that (Ω, A, µ) is a complete probability space, (X, ∥ • ∥) is a real separable Banach space, K k (X) is the family of all nonempty compact subsets of X, and K kc (X) is the family of all nonempty compact convex subsets of X.
Let A and B be two nonempty subsets of X and let λ ∈ R, the set of all real numbers.We define addition and scalar multiplication by The Hausdorff metric on K k (X) is defined by The metric space (K k (X), d H ) is complete and separable, and K kc (X) is a closed subset of (K k (X), d H ) (cf. (Li, Ogura & Kreinovich, 2002), Theorems 1.1.2and 1.1.3).Concerning the concepts and results of set-valued random variables, readers may refer to the book (Li, Ogura & Kreinovich, 2002).
A set-valued mapping In fact, set-valued random variables can be defined as mappings from Ω to the family of all closed subsets of X. Concerning its equivalent definitions, please refer to (Castaing & Valadier, 1977) and (Hiai & Umegaki, 1977).

e.(µ).
For each set-valued random variable F, define the family of integrable X-valued random variables.This integral was first introduced by Aumann (Aumann, 1965), called Aumann integral in literature.
For each A ∈ K(X), define the support function by where X * is the dual space of X.
Let S * denote the unit sphere of X * , C(S * ) the all continuous functions of S * , and the norm is defined as ∥v∥ C = sup x * ∈S * From now on, we begin to introduce necessary concepts, notation and basic results on fuzzy set space and fuzzy set-valued random variables.
Let F k (X) be the family of all special fuzzy sets: v : X → [0, 1] satisfying the following conditions: The uniform metric in F k (X), which is an extension of the Hausdorff metric d H , is often used (cf.Puri & Ralescu, 1986) , where I 0 is the function taking value one at 0 and zero for all x 0. The space (F k (X), d ∞ H ) is a complete metric space (cf.Li, & Ogura, 1996) but not separable (Li, Ogura & Kreinovich, 2002), Remark 5.1.7).Completeness was first proved by Puri and Ralescu (Puri & Ralescu, 1986) in the case of X = R d , the d-dimensional Euclidean space.
A fuzzy set-valued random variable (or F k (X)-valued random variable) is a measurable mapping X from the space (Ω, A) ) , which will be used later.Obviously, v 0+ is the support set of v. Due to the completeness of (F k (X), d ∞ H ), every Cauchy sequence {v n : n ∈ N} has a limit v in F k (X).
A sequence of set-valued random variables {F n : n ∈ N} is called to be stochastically dominated by a set-valued random variable F if A sequence of fuzzy set-valued random variables {X n : n ≥ 1} is called to be stochastically dominated by a fuzzy setvalued random variable X if It is obvious that if {X n : n ≥ 1} is stochastically dominated by a fuzzy set-valued random variable X, then ∀ α ∈ (0, 1], {X n α : n ≥ 1} is stochastically dominated by the set-valued random variable X α .And also for any α ∈ (0, 1], {X n α+ : n ≥ 1} is stochastically dominated by the set-valued random variable X α+ (Guan & Li, 2004).

Negatively Dependent and Main Results
In this section, we will give the definition of negatively dependent for fuzzy set-valued random variables and discuss the properties.Then we will prove the weak and strong limit theorem of weighted sums for fuzzy set-valued negatively dependent random variables in the sense of d ∞ H .The following definition is Toeplitz sequence, which will be used later.

Definition 3.1 A double array {a
Now we will recall some concepts of negatively dependent random variables.Definition 3.2 (cf.Bozorgnia, Patterson & Taylor, 1993)) A finite family of real-valued random variables X 1 , • • • , X n is said to be negatively dependent if for all real x 1 , x 2 , • • • , x n , An infinite family of random variables is negatively dependent if every finite subfamily is negatively dependent.The following results are from (Bozorgnia, Patterson & Taylor, 1993) of Bozorgnia et. al., and we will use them in the later.
Lemma 3.3 (cf.Bozorgnia, Patterson & Taylor, 1993) Let real-valued random variables {X i : 1 ≤ i ≤ n} be negatively dependent.Then the following are true: be all nondecreasing (or all nonincreasing) Borel functions, then random variables g 1 (X 1 ), g 2 (X 2 ), Now we give the definition of fuzzy set-valued random variables and discuss the property.Definition 3.5 A finite family of fuzzy set-valued random variables X 1 , X 2 , • • • , X n be said to be negatively dependent if for any α ∈ (0, 1], the set-valued random variables X 1 α , X 2 α , • • • , X n α are negatively dependent.
From the definition 3.5, we can easily obtain the following result.
Then by the continuous of probability and lemma 3.3, we have Then the result was proved. 2 The following limit theorem is a weak convergence result for fuzzy set-valued negatively dependent random variables, which is the extension of (Guan & Sun, 2014).Here we assume the Banach space X = R.
Proof.From the definition 3.5 and theorem 3.6, we know that {X nk α }, {X nk α+ } ∈ K k (R + ) are all rowwise negatively dependent set-valued random variables.Then by theorem 4.1 of (Guan & Sun, 2014), for any α ∈ (0, 1], ε > 0, we have and Take a finite partition 0 , by virtue of monotone property of level sets and the formula ) .
Then we have The result was proved. 2 Next, we shall prove the strong convergence theorem, which is the extend of theorem 4.1 of (Guan & Wan, 2016).In (Guan & Wan, 2016), the authors proved the convergence theorem of set-valued random variables in the sense of d H .Here we extend their result to fuzzy set-valued random variables, and the metric is d ∞ H . Theorem 3.8 Let {X nk } ∈ F k (R + ) be an array of rowwise negatively dependent fuzzy set-valued random variables with E[X nk ] = I 0 and stochastic dominated by a fuzzy set-valued random variable X.If max We can find a finite partition 0 ) ) .
Since the two terms on the right hand converge to 0 in the sense of d H , then we can have The result was proved. 2 From theorem 3.8, we can easily get the following two corollaries.
Corollary 3.9 If {X n } is a sequence of independent identical distributed fuzzy set-valued random variables in F k (R + ) with E[X 1 ] = I 0 and max a nk X nk → 0 a.e.
Theorem 3.7 Let {X nk : k ≥ 1, n ≥ 1} be an array of fuzzy set-valued random variables in F k (R + ) which are stochastically dominated by a fuzzy set-valued random variable X, and are pairwise negatively dependent in each row.Let E[X nk ] = I 0 for all n and k.Let {a nk } be an array of nonnegative real numbers such that (Guan & Wan, 2016)n 3.5 and theorem 3.6, we know that{X nk α }, {X nk α+ } ∈ K k (R +) are all rowwise negatively dependent set-valued random variables.Then by theorem 4.1 of(Guan & Wan, 2016), we have with respect to the Hausdorff metric d H .And ∞ ∑ k=1 a nk X nk α+ → 0 a.e. with respect to the Hausdorff metric d H .
By virtue of monotone property of level sets and the formula ( ∞ ∑ k=1 a nk X nk ) α =