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In this paper, we obtain some multivalued strong laws of large numbers for triangular array of rowwise exchangeable random sets and fuzzy random sets in a separable Banach space in the Mosco sense. Our results are obtained without bounded expectation condition, with or without compactly uniformly integrable and reverse martingale hypotheses. They improve some related results in literature. Some typical examples illustrating this study are provided
slln for triangular array of row-wise exchangeable random sets and fuzzy random sets with respect to mosco convergence Nguyen Van Quang∗, Duong Xuan Giap† Abstract In this paper, we obtain some multivalued strong laws of large numbers for triangular array of row-wise exchangeable random sets and fuzzy random sets in a separable Banach space in the Mosco sense. Our results are obtained without bounded expectation condition, with or without compactly uniformly integrable and reverse martingale hypotheses. They improve some related results in literature. Some typical examples illustrating this study are provided. Mathematics Subject Classifications (2010): 60F15, 60B12, 28B20. Key words and phrases: triangular array, random set, strong law of large numbers, Mosco convergence, exchangeability. 1 Introduction In recent decades, the strong laws of large numbers (SLLN) for unbounded random sets, gave rise to applications in several fields, such as optimization and control, stochastic and integral geometry, mathematical economics, statistics and related fields. The first multivalued SLLN was proved by Artstein and Vitale [1] for independent identically distributed (i.i.d.) random variables whose values are compact subsets of Rd . Puri and Ralescu [19] were the first to obtain the SLLN for i.i.d. Banach space-valued compact convex random sets. Later, Hiai [8] and Hess [6] independently proved similar results for random sets in an infinite dimensional Banach space, with respect to the Mosco convergence. Further variants of the multivalued SLLN have been established under various conditions, for example, see Castaing, Quang and Giap [2, 3], Fu and Zhang [4, 5], Inoue [12, 13], Kim [15], Quang and Giap [21, 22], Quang and Thuan [23]. Moreover, Hess [7] established the Mosco convergence of multivalued supermartingales and supermartingale integrands. Later, Li and Ogura [11] proved the convergence theorems of set-valued and fuzzy-valued martingales in the Mosco sense without assuming that their values are compact or of compact level sets. They also obtained some convergence theorems of closed and convex set valued sub- and supermartingales in the Mosco topology (see Li and Ogura [10]). The first result on multivalued SLLN with respect to Mosco convergence for triangular array of random sets was established by Quang and Giap [21]. In this paper, the authors established the SLLN for triangular array of row-wise independent random sets in Banach space with bounded expectation condition. According to this direction, in present paper, we study the Mosco convergence of the SLLN for triangular array of row-wise exchangeable random sets. However, in [21], the SLLN was established under the bounded expectation condition, while in the present paper, this condition is not assumed. To give the main results, we provide a new method in building structure of triangular array of selections to prove the “lim inf” path of Mosco convergence. We also use a condition of the Mosco convergence in the first column of triangular array of random sets and fuzzy random sets, which was introduced ∗ Department † Department of Mathematics, Vinh University, Nghe An Province, Viet Nam. Email: nvquang@hotmail.com of Mathematics, Vinh University, Nghe An Province, Viet Nam. Email: dxgiap@gmail.com 1 by Hiai [8] and which was also used by other authors. Our results improve some related results in literature. The organization of this paper is as follows: In Section 2, we introduce some basic notions: setvalued random variable, fuzzy-valued random variable, Mosco convergence and exchangeability. Section 3 is concerned with some theorems on Mosco convergence of the SLLN for triangular arrays of row-wise exchangeable random sets and fuzzy random sets in a separable Banach space. A new method in building structure of triangular array of selections to prove the “lim inf” path of the Mosco convergence is provided. Illustrative examples are also provided in this section. 2 Preliminaries Throughout this paper, let (Ω, F, P) be a complete probability space, (X, . ) be a real separable Banach space and X∗ be its topological dual. The σ-field of all Borel sets of X is denoted by B(X). In the present paper, R (resp. N) will be denoted the set of real numbers (resp. the set of positive integers). Let c(X) be the family of all nonempty closed subsets of X and E(X) (shortly, E) be the Effros σ-field on c(X). This σ-field is generated by the subsets U − = {F ∈ c(X) : F ∩ U = ∅}, where U ranges over the open subsets of X. On the other hand, for each A, C ⊂ X, clC, coC and coC denote the norm-closure, the convex hull and the closed convex hull of C, respectively; the distance function d(·, C) of C, the Hausdorff distance dH (A, C) of A and C, the norm C of C and the support function s(C, ·) of C are defined by d(x, C) = inf{ x − y : y ∈ C}, (x ∈ X), dH (A, C) = max{sup d(x, C), sup d(y, A)}, x∈A y∈C C = dH (C, {0}) = sup{||x|| : x ∈ C}, s(C, x∗ ) = sup{ x, x∗ : x ∈ C}, (x∗ ∈ X∗ ). The space c(X) has a linear structure induced by Minkowski addition and scalar multiplication: A + B = {a + b : a ∈ A, b ∈ B}, λA = {λa : a ∈ A}, where A, B ∈ c(X), λ ∈ R. A multivalued (set-valued) function X: Ω → c(X) is said to be F-measurable (or measurable) if X is (F, E)−measurable, i.e., for every open set U of X, the subset X −1 (U − ) = {ω ∈ Ω : X(ω) ∩ U = ∅} belongs to F. A measurable multivalued function is also called a closed valued random variable (or random set). The sub-σ-field X −1 (E) generated by X is denoted by FX . The distribution PX of the random set X : Ω → c(X) on the measurable space (c(X), E) is defined by PX (B) = P{X −1 (B)}, for all B ∈ E. A collection of random sets {Xi , i ∈ I} is said to be identically distributed (i.d.) if the PXi , i ∈ I are identical. A random element (Banach space valued random variable) f : Ω → X is called a selection of the random set X if f (ω) ∈ X(ω) for all ω ∈ Ω. For every sub-σ-field A of F and for 1 ≤ p < ∞, Lp (Ω, A, P, X) denotes the Banach space of 1 (equivalence classes of) measurable functions f : Ω → X such that the norm f p = (E f p ) p = 1 f (ω) p dP p is finite. In special case, Lp (Ω, F, P, X) (resp. Lp (Ω, F, P, R)) is denoted by Lp (X) (resp. Lp ). For each random set X, define the following closed subset of Lp (Ω, A, P, X) Ω p SX (A) = {f ∈ Lp (Ω, A, P, X) : f (ω) ∈ X(ω), for all ω ∈ Ω}. 1 A random set X : Ω → c(X) is called integrable if the set SX (F) is nonempty (i.e. d(0, X(·)) is in L ), and it is called integrable bounded if the random variable X is in L1 . For any random set X and any sub-σ-field A of F, the multivalued expectation of X over Ω, with respect to A, is defined by 1 E(X, A) = {E(f ) : f ∈ SX (A)}, 1 2 where E(f ) = Ω f dP is the usual Bochner integral of f . Shortly, E(X, F) is denoted by EX. We note that E(X, A) is not always closed. The sequence of random elements {Xn : n ≥ 1} is called a martingale sequence if E Xn < ∞ and Xn = E(Xn+m |X1 , X2 , . . . , Xn ) a.s. for all positive integers m and n. Similarly, {Xn : n ≥ 1} is called a reverse martingale sequence if it is a martingale under the reverse ordering of N, that is, Xm+n = E(Xn |Xm+n , Xm+n+1 , . . . ) a.s. for all positive integers m and n. A sequence of random elements {Xn : n ≥ 1} is said to be tight if for each > 0 there exists a compact subset K of X such that P[Xn ∈ / K ] < for every positive integer n. Also, a general condition involving tightness of distributions and moments of the random elements {Xn : n ≥ 1} called compact uniform integrability (CUI) can be stated as: Given > 0, there exists a compact subset K of X such that supn (E Xn I[Xn ∈K / ] ) < , where IA is the indicator function of A. Next, we describe some basic concepts of fuzzy random sets. A fuzzy set in X is a function u : X → [0, 1]. For each fuzzy set u, the α-level set is denoted by Lα u = {x ∈ X : u(x) ≥ α}, 0 < α ≤ 1. It is easy to see that, for every α ∈ (0, 1], Lα u = ∩β 0, by [2, Lemma 3.6], we can choose x1 , x2 , ..., xm ∈ X 6 (the elements x1 , x2 , ..., xm only depend on x and ) such that m 1 m xj − x < . j=1 Therefore, we only need to show that there exists a triangular array {fni : n ≥ 1, 1 ≤ i ≤ n} of selections of {Xni } such that 1 n n fni (ω) → i=1 1 m m xj a.s. as n → ∞. (3.4) j=1 m 1 Indeed, let zm = m j=1 xj . The statement (3.4) means that zm ∈ s-liGn (ω) a.s. Since the space X is separable, there exists a countable dense set DcoX of coX. For each fixed x(j) ∈ DcoX and for every k = k1 (k ≥ 1), by (3.4), there exists a positive integer mk , which depends on x(j) and k , such that zmk ∈ s-liGn (ω) a.s. Therefore, there exists Nk ∈ F such that P(Nk ) = 1 and ∞ zmk ∈ s-liGn (ω) for all ω ∈ Nk . Let N = k=1 Nk , then P(N ) = 1. For each ω ∈ N , it follows from the set s-liGn (ω) is closed, zmk ∈ s-liGn (ω) for all k and zmk → x(j) as k → ∞, that x(j) ∈ s-liGn (ω). This means that x(j) ∈ s-liGn (ω) a.s., for each j ≥ 1. Noting that DcoX is a countable set, we obtain DcoX ⊂ s-liGn (ω) a.s. Since the set s-liGn (ω) is closed for each ω, by taking the closure of both sides of the above relation, we have coX ⊂ s-liGn (ω) a.s. Therefore, the statement (3.4) is proved. (j) (j) 1 By (3.2), for each j ∈ {1, 2, . . . , m}, there exists a sequence {gn1 : n ≥ 1} of gn1 ∈ SX (FXn1 ) n1 (j) (j) (j) (j) L (j) such that gn1 − Egn1 ≥ g(n+1),1 − Eg(n+1),1 for each n and gn1 →2 xj as n → ∞. Since {Xni : n ≥ 1, 1 ≤ i ≤ n} is row-wise exchangeable and by virtue of Lemma 3.1(2), it follows (j) that for each j ∈ {1, 2, . . . , m} and for each n ≥ 1, there exists a sequence {gni : 1 ≤ i ≤ n} of (j) (j) 1 gni ∈ SX (FXni ) such that the sequence {gni : 1 ≤ i ≤ n} is exchangeable. By Lemma 3.6 for the ni (j) case of single-valued random variables, we get E gni − xj 2 (j) = E gn1 − xj 2 for all i ∈ {1, 2, . . . , n}. (j) L gni →2 xj as n → ∞ for each i and j. It follows that 1 (FXni ) satisfying Now, we will construct a triangular array {fni : n ≥ 1, 1 ≤ i ≤ n} of fni ∈ SX ni (3.4) as follows: (j) fni (ω) := gni (ω) if i ≡ j (mod m), where j ∈ {1, 2, . . . , m} and for all ω ∈ Ω. (3.5) This means that fni n≥1,1≤i≤n (1) g11 (1) g21 .. . g (1) m,1 = (1) gm+1,1 .. . 1st column (1) (2) g22 .. . .. (2) gm,2 ... gm,m (2) ... gm+1,m .. . gm+1,m+1 .. . mth column (m) of {gni } (m+1)th column (1) of {gni } gm+1,2 .. . 2 of {gni } . ... nd column (2) of {gni } (m) (m) (1) .. . Then, for each m ≥ 1 and j ∈ {1, 2, . . . , m}, the array {fn,(i−1)m+j } is row-wise exchangeable and L fn, (i−1)m+j →2 xj as n → ∞, for each i ≥ 1. (Let us note that {fn,(i−1)m+j } is not a triangular array of random elements). 7 (3.6) Let yni = Efni , n ≥ 1, 1 ≤ i ≤ n. If n = (k − 1)m + l, where 1 ≤ l ≤ m, then the following estimations hold: 1 n n fni (ω) − i=1 = ≤ 1 n k n m 1 m m xj j=1 k fn,(i−1)m+j (ω) − j=1 i=1 m j=1 1 n m fn,(k−1)m+j (ω) − j=l+1 k 1 k (fn,(i−1)m+j (ω) − xj ) + i=1 k n m j=1 + fn,(k−1)m+j (ω) (fn,(i−1)m+j (ω) − yn,(i−1)m+j ) + i=1 1 n m xj j=1 k n m j=1 m fn,(k−1)m+j (ω) − yn,(k−1)m+j + j=l+1 + xj j=1 j=l+1 k 1 k m m 1 k − n m + ≤ 1 n 1 m k 1 − n m 1 n 1 k k yn,(i−1)m+j − xj i=1 m yn,(k−1)m+j j=l+1 m xj . (3.7) j=1 Let gni (ω) = fni (ω) − yni , for all ω ∈ Ω, n ≥ 1 and 1 ≤ i ≤ n. By Lemma 3.6 for the case of singlevalued random variables, we get that if a sequence {fk : k ≥ 1} of random elements is exchangeable then the sequence {fk + c : k ≥ 1} is exchangeable, too (where c is a constant in X). Therefore, since the array {fn,(i−1)m+j : n ≥ 1, 1 ≤ i ≤ k} of random elements is row-wise exchangeable, we obtain Efn,(i−1)m+j = c for all i (here, n and j are fixed). From the above statements, we deduce that the array {gn,(i−1)m+j : n ≥ 1, 1 ≤ i ≤ k} is row-wise exchangeable, too. L By (3.6), for each s = (i − 1)m + j (1 ≤ j ≤ m), we have fns →2 xj as n → ∞; namely, E fns − xj 2 → 0 as n → ∞, and so 0 ≤ Efns − xj 2 = E(fns − xj ) 2 2 ≤ (E fns − xj ) ≤ E fns − xj 2 (by EX ≤ E X ) → 0 as n → ∞. (by the inequality for convex function) Since then, we get 0 ≤ gns 2 = (fns − xj ) − (Efns − xj ) 2 ≤ fns − xj 2 + Efns − xj n→∞ 2 → 0. This means that L gns →2 0 as n → ∞. (3.8) Further, for each f ∈ X∗ , we have ρn (f ) = E (f (gni )f (gnj )) = E (f (fni − Efni ).f (fnj − Efnj )) = E [(f (fni ) − f (Efni )).(f (fnj ) − f (Efnj ))] (by f is a linear mapping) = E [(f (fni ) − E(f (fni ))).(f (fnj ) − E(f (fnj )))] (by the definition of expectation of random elements) = Cov (f (fni ), f (fnj )) = Cov (f (gn (coXn1 )), f (gn (coXn2 ))) → 0 as n → ∞ for all i = j and i ≡ j (mod m). (by (3.1)) 8 (3.9) (j) For each n = (k − 1)m + l, set Sn (ω) = 1 k k i=1 gn,(i−1)m+j (ω) for all ω ∈ Ω. For each (j) {Sn j ∈ {1, 2, . . . , m}, the sequence : n ≥ 1} of random elements is divided into m subsequences (j) {S(k−1)m+l : k ≥ 1}, l ∈ {1, 2, . . . , m}. From the above statements, the triangular array {g(k−1)m+l,(i−1)m+j : k ≥ 1, 1 ≤ i ≤ k} satisfies all the conditions of Lemma 3.2 for each l, j ∈ {1, 2, . . . , m}. Applying this lemma, we obtain (j) S(k−1)m+l (ω) = 1 k k g(k−1)m+l,(i−1)m+j (ω) → 0 a.s. as k → ∞, (3.10) i=1 for each l, j ∈ {1, 2, . . . , m}. It is equivalent to 1 k Sn(j) (ω) = k gn,(i−1)m+j (ω) → 0 a.s. as n → ∞, for each j ∈ {1, 2, . . . , m}. (3.11) i=1 (j) For each n ≥ 1 and j ∈ {1, 2, . . . , m}, we set Vn 1 k = (j) {Vn : n ≥ 1} of real numbers is divided into m subsequences For each l, j ∈ {1, 2, . . . , m}, we put (l,j) zki k i=1 yn,(i−1)m+j − xj . (j) {V(k−1)m+l : k ≥ 1}, l ∈ The sequence {1, 2, . . . , m}. = y(k−1)m+l,(i−1)m+j − xj . For each j ∈ {1, 2, . . . , m}, by the assumption that the array {fn,(i−1)m+j : n ≥ 1, 1 ≤ i ≤ k} is row-wise exchangeable and converges in the second mean to xj as n → ∞ for each column, we get that the elements of this array have bounded expectations. Therefore, (l,j) |zki | ≤ Ef(k−1)m+l,(i−1)m+j + xj ≤ C + xj , (3.12) for all k ≥ 1, 1 ≤ i ≤ k. (l,j) Since the convergence in L2 implies the convergence in L1 and by (3.6), we have that zki → 0 as (l,j) k → ∞, for each i ≥ 1. Since then, zki → 0 as i → ∞. (l,j) Combining this with (3.12), we have that for each l, j ∈ {1, 2, . . . , m}, the triangular array {zki : k ≥ 1, 1 ≤ i ≤ k} of real numbers satisfies all the conditions of Lemma 3.5. Applying this lemma, we obtain (j) V(k−1)m+l = 1 k k y(k−1)m+l,(i−1)m+j − xj → 0 as k → ∞ for each l, j ∈ {1, 2, . . . , m}. i=1 Hence, Vn(j) 1 = k k yn,(i−1)m+j − xj → 0 as n → ∞. (3.13) i=1 By (3.11), we have 1 n m fn,(k−1)m+j (ω) − yn,(k−1)m+j = j=l+1 = k n m j=l+1 1 k k gn,(i−1)m+j (ω) − ( i=1 1 n m gn,(k−1)m+j (ω) j=l+1 k−1 1 ) k k−1 9 k−1 gn,(i−1)m+j (ω) → 0 as n → ∞. i=1 (3.14) Similarly, by (3.13), we obtain 1 n m yn,(k−1)m+j ≤ j=l+1 = m k n j=l+1 1 k 1 n m yn,(k−1)m+j − xj + j=l+1 k yn,(i−1)m+j − xj − ( i=1 k−1 1 ) k k−1 + 1 n 1 n m xj j=l+1 k−1 yn,(i−1)m+j − xj i=1 m xj → 0 as n → ∞. (3.15) j=l+1 1 We also have ( nk − m ) → 0 as n → ∞. Therefore, combining (3.7), (3.11), (3.13), (3.14) and (3.15), we get n m 1 1 fni (ω) − xj → 0 a.s. as n → ∞. n i=1 m j=1 This yields 1 m m j=1 xj ∈ s-liGn (ω) a.s. Hence coX ⊂ s-liGn (ω) a.s. Thus, in the above proving, the triangular array {gni : n ≥ 1, 1 ≤ i ≤ n} of random elements has been divided into m2 triangular sub-arrays {g(k−1)m+l,(i−1)m+j : k ≥ 1, 1 ≤ i ≤ k}. Also, for each j ∈ {1, 2, . . . , m}, the array { yn,(i−1)m+j − xj : n ≥ 1, 1 ≤ i ≤ k} of real numbers has been divided (l,j) into m triangular sub-arrays {zki : k ≥ 1, 1 ≤ i ≤ k}, l ∈ {1, 2, . . . , m}. By using Lemma 3.2 (resp. Lemma 3.5) for each above triangular sub-array of random elements (resp. of real numbers), we obtain the “lim inf” path of the Mosco convergence. Next, let {xj : j ≥ 1} be a dense sequence of X \ coX. By the separation theorem, there exists a sequence {x∗j : j ≥ 1} in X∗ with x∗j = 1 such that xj , x∗j − d(xj , coX) ≥ s(coX, x∗j ), for every j ≥ 1. (3.16) Then x ∈ coX if and only if x, x∗j ≤ s(coX, x∗j ) for every j ≥ 1. Note that the function X → s(X, x∗j ) of c(X) into (−∞, ∞] is (E, B(R))-measurable. Using the above statement, the inequality (3.16), the hypotheses of this theorem and Lemma 3.6, we have that {s(Xni , x∗j ) : n ≥ 1, 1 ≤ i ≤ n} is a triangular array of row-wise exchangeable random (j) (j) variables in L1 , for each j ≥ 1. Set hni = s(Xni , x∗j ) − E(s(Xni , x∗j )). Then, {hni : n ≥ 1, 1 ≤ i ≤ n} is the triangular array of row-wise exchangeable random variables. (j) L By the condition (3.3), using the arguments as in the proof of (3.8), we get hn1 →2 0 as n → ∞. It (j) L implies that hni →2 0 as n → ∞, for each i ≥ 1. (j) (j) By the condition (3.1), we have that ρn (f ) = E(hni hnk ) → 0 as n → ∞ for all i = k. (j) From the above statements, we get that the triangular array {hni : n ≥ 1, 1 ≤ i ≤ n} satisfies all the conditions of Lemma 3.2 for real-valued random variables, for each j ≥ 1. Then, applying this lemma, we have n 1 (j) h (ω) → 0 a.s. as n → ∞, for every j ≥ 1. n i=1 ni This means that 1 n n s(Xni , x∗j ) − i=1 1 n n E(s(Xni , x∗j )) → 0 a.s. as n → ∞, for every j ≥ 1. i=1 Moreover, by (3.3) and (3.16), we get Es(Xni , x∗j ) = s(clE(Xni ), x∗j ) → s(X, x∗j ) < ∞ as n → ∞ for every i, j ≥ 1. Therefore, for each i and j, the sequence {s(Xni , x∗j ) : n ≥ 1} has bounded expectation. 10 Since Es(Xni , x∗j ) = Es(Xn1 , x∗j ) for all i ∈ {1, 2, . . . , n}, the triangular array {s(Xni , x∗j ) : n ≥ 1, 1 ≤ i ≤ n} has bounded expectation. Since then, by applying Lemma 3.5, we have 1 n n E(s(Xni , x∗j )) → s(X, x∗j ) as n → ∞, for every j ≥ 1. i=1 Consequently, for each j ≥ 1, s(Gn (ω), x∗j ) → s(X, x∗j ) a.s. as n → ∞. Namely, there exists N ∈ F, P(N ) = 0 such that for each ω ∈ Ω\N, j ≥ 1, s(Gn (ω), x∗j ) → s(X, x∗j ) as n → ∞. w For each ω ∈ Ω\N, if x ∈ w-lsGn (ω) then xk → x as k → ∞, where xk ∈ Gn(k) (ω). Hence, x, x∗j = lim xk , x∗j ≤ lim s(Gn(k) (ω), x∗j ) = s(X, x∗j ) = s(coX, x∗j ), for every j ≥ 1. k→∞ k→∞ This implies that x ∈ coX. Thus, w-lsn→∞ n1 cl n i=1 Xni (ω) ⊂ coX a.s. By putting Xni (ω) = Xi (ω) for every n ≥ 1, 1 ≤ i ≤ n and ω ∈ Ω, and applying Theorem 3.7 for the triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n}, we get Corollary 3.8. ( Inoue and Taylor [14, Theorem 4.3]) Let {Xn : n ≥ 1} be an infinite sequence of exchangeable random sets in c(X). If E X1 < ∞ and Cov{f (g(coX1 )), f (g(coX2 ))} = 0 for each f ∈ X∗ , then n 1 Xk → coEX1 in Mosco sense, n k=1 where g ∈ I1 (coX1 , coX2 ). Theorem 3.9. Let {Xni : n ≥ 1, 1 ≤ i ≤ n} be a triangular array of row-wise exchangeable random sets in separable Banach space X. 1 (FXni ) such that Assume that there exists a triangular array {fni : n ≥ 1, 1 ≤ i ≤ n} of fni ∈ SX ni L {fni } is row-wise exchangeable, fnk →2 f∞k as n → ∞ for each k ≥ 1 and fn1 − f∞1 ≥ f(n+1),1 − f∞1 for all n. (3.17) Suppose that there exists a nonempty subset X of X such that: +) For each x ∈ X, E [f (gn1 (coXn1 ) − x).f (gn2 (coXn2 ) − x)] → 0 as n → ∞, for all f ∈ X∗ (3.18) and gni ∈ I1 (coXni ), n ≥ 1, i ∈ {1, 2}. (x∗ ) L +) For each x∗ ∈ X∗ and k ≥ 1, s(Xnk , x∗ ) →2 S∞k as n → ∞, and (x∗ ) (x∗ ) |s(Xn1 , x∗ ) − S∞1 | ≥ |s(X(n+1),1 , x∗ ) − S∞1 | for all n, then n M - lim 1 cl Xni (ω) = coX a.s. n i=1 Proof. By the arguments as in the proof of Theorem 3.7, for each x ∈ coX and x1 , x2 , ..., xm ∈ X such that m 1 xj − x < . m j=1 1 m (3.19) > 0, there exist To prove the “lim inf” path coX ⊂ s-liGn (ω) a.s. in the Mosco convergence, we need to show that m j=1 xj ∈ s-liGn (ω) a.s. 11 If n = (k − 1)m + l, where 1 ≤ l ≤ m, then the following estimations hold: 1 n n fni (ω) − i=1 1 n = k n ≤ m 1 m m xj j=1 k fn,(i−1)m+j (ω) − j=1 i=1 m 1 k j=1 1 n m fn,(k−1)m+j (ω) − j=l+1 k (fn,(i−1)m+j (ω) − xj ) + i=1 + 1 n 1 m m xj j=1 m fn,(k−1)m+j (ω) j=l+1 k 1 − n m m xj . (3.20) j=1 By (3.18), we have ρn (f, xj ) = E[f (fns − xj ).f (fnk − xj )] → 0 as n → ∞, (3.21) for all s = k, f ∈ X∗ , for each j ∈ {1, 2, . . . , m}. By the arguments as in the proof of Theorem 3.7, we have that for each j ∈ {1, 2, . . . , m}, the rowwise exchangeability of array {fn,(i−1)m+j } of random elements implies the row-wise exchangeability of array {fn,(i−1)m+j − xj }. (j) k For each n = (k − 1)m + l, we put Sn (ω) = k1 i=1 (fn,(i−1)m+j (ω) − xj ) for all ω ∈ Ω. For each (j) j ∈ {1, 2, . . . , m}, the sequence {Sn : n ≥ 1} of random elements is divided into m subsequences (j) {S(k−1)m+l : k ≥ 1}, l ∈ {1, 2, . . . , m}. L By (3.17), we get that fn,(i−1)m+j − xj →2 f∞,(i−1)m+j − xj as n → ∞ for each i ≥ 1, j ∈ {1, 2, . . . , m} and (fn,(i−1)m+j − xj ) − (f∞,(i−1)m+j − xj ) = fn,(i−1)m+j − f∞,(i−1)m+j . Therefore, the triangular array {f(k−1)m+l,(i−1)m+j − xj : k ≥ 1, 1 ≤ i ≤ k} of random elements satisfies all the conditions of Lemma 3.2 for each l, j ∈ {1, 2, . . . , m} and we have (j) S(k−1)m+l (ω) = 1 k k (f(k−1)m+l,(i−1)m+j (ω) − xj ) → 0 a.s. as k → ∞, (3.22) i=1 for each l, j ∈ {1, 2, . . . , m}. It implies that Sn(j) (ω) → 0 a.s. as n → ∞, for each j ∈ {1, 2, . . . , m}. (3.23) Since n → ∞ implies k → ∞, by (3.23), we obtain 1 n m fn,(k−1)m+j (ω) j=l+1 = k n m j=l+1 1 k k (fn,(i−1)m+j (ω) − xj ) − ( i=1 k−1 1 ) k k−1 k−1 (fn,(i−1)m+j (ω) − xj ) + i=1 → 0 as n → ∞. Since then, combining (3.20), (3.23) and (3.24), we have 1 n Hence, 1 m m j=1 n fni (ω) − i=1 1 m m xj → 0 a.s. as n → ∞. j=1 xj ∈ s-liGn (ω) a.s. 12 1 xj k (3.24) Let {x∗j : j ≥ 1} be as in the proof of Theorem 3.7 taken for coX. To prove the “lim sup” path n w-ls n1 cl i=1 Xni (ω) ⊂ coX a.s. in the Mosco convergence, we argue as in the proof of Theorem 3.7. (j) Detail, for each j ≥ 1, set hni (ω) = s(Xni (ω), x∗j ) − s(X, x∗j ). By using Lemma 3.2, we obtain 1 n n (j) hni (ω) → 0 a.s. as n → ∞, for each j ≥ 1. i=1 This means that 1 n n s(Xni , x∗j ) − s(X, x∗j ) → 0 a.s. as n → ∞, for each j ≥ 1. i=1 It is equivalent to s(Gn (ω), x∗j ) → s(X, x∗j ) a.s., for each j ≥ 1. Thus, we obtain the desired conclusion. Remark. Let us note that the conclusion of Theorem 3.9 will be only coX ⊂ s-liGn (ω) a.s., if the condition (3.18) is not assumed. At this point, Theorem 3.9 extends the result of Taylor and Patterson (1985, Theorem 1) for multivalued random variables. Indeed, suppose that the triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n} of random elements in a separable Banach space satisfies all the conditions of Lemma 3.2. We can check that the triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n} satisfies all the conditions of Theorem 3.9 without the condition (3.19) for single-valued random variables case with X = {0}. By using Theorem 3.9, we obtain the SLLN as in Lemma 3.2. Next, we will establish a multivalued SLLN for triangular array of row-wise exchangeable random sets with CUI and reverse martingale conditions. To do this, we need the following lemma. Lemma 3.10. Let {fn : n ≥ 1} be a sequence of random elements in L1 (X). Suppose that the sequence {fn : n ≥ 1} is CUI and Efn → x as n → ∞, where x is an element of X. Then, the sequence {fn − Efn : n ≥ 1} is CUI. Proof. Given > 0, there exists a compact subset K1 of X such that sup E fn I[fn ∈K / 1] < n 2 . (3.25) We put K2 = cl{Efn | n ≥ 1}. By the convergence of the sequence {Efn : n ≥ 1}, we have that K2 is a compact subset of X. We set K = K1 − K2 , then K is a compact subset. Now, we will show that K1 ⊂ Efn + K, for every n ≥ 1. (3.26) Indeed, for each k1 ∈ K1 , it follows from K = K1 − K2 and Efn ∈ K2 that kn = k1 − Efn ∈ K. This yields k1 = Efn + kn ∈ Efn + K. Thus , (3.26) is proved. By (3.26), we get [fn ∈ / Efn + K] ⊂ [fn ∈ / K1 ], for every n ≥ 1. Next, we have that E (fn − Efn )I[fn −Efn ∈K] ≤ 2E fn I[fn −Efn ∈K] / / = 2E fn I[fn ∈Ef / n +K] ≤ 2E fn I[fn ∈K / 1 ] , for every n ≥ 1 (by (3.27)). By (3.25), we obtain sup E (fn − Efn )I[fn −Efn ∈K] < . / n The lemma is proved completely. 13 (3.27) Theorem 3.11. Let {Xni : n ≥ 1, 1 ≤ i ≤ n} be a triangular array of row-wise exchangeable random sets in the separable Banach space X. If for every f ∈ X∗ , +) the sequence {gn (Xn1 ) : n ≥ 1} is CUI, with gn ∈ I1 (coXn1 ), (3.28) +) {E(gn (coXn1 )|G(n, m, j)) : n ≥ 1} is a reverse martingale, for each m ≥ 1, j ∈ {1, 2, . . . , m}, where I(n, m, j) = {(k − 1)m + j|1 ≤ (k − 1)m + j ≤ n, k ∈ N}, gn ∈ I1 (coXn1 ) gn+1 (coXn+1,k ), . . . }, gn (coXnk ), and G(n, m, j) = σ{ k∈I(n,m,j) (3.29) k∈I(n+1,m,j) +) Cov (f (gn (coXn1 )), f (gn (coXn2 ))) → 0 as n → ∞, with gn ∈ I1 (coXn1 , coXn2 ), (3.30) +) V ar(f (gn (coXn1 ))) = o(n), with gn ∈ I1 (coXn1 ), (3.31) +) there exists a set X ∈ c(X) such that X ⊂ s-liclE(Xn1 , FXn1 ), (3.32) ∗ ∗ ∗ ∗ lim sup s(clEXn1 , x ) ≤ s(X, x ) for all x ∈ X , then (3.33) n M - lim 1 cl Xni (ω) = coX a.s. n i=1 Proof. As in the proof of Theorem 3.7, to prove the “lim inf” path in the Mosco convergence, we need 1 (F) such that to show that there exists a triangular array {fni : n ≥ 1, 1 ≤ i ≤ n} of fni ∈ SX ni 1 n n fni (ω) → i=1 1 m m xj a.s. as n → ∞. j=1 (j) (j) 1 By (3.32), for each j ∈ {1, 2, . . . , m}, there exists a sequence {gn1 : n ≥ 1} of gn1 ∈ SX (FXn1 ) n1 (j) such that Egn1 → xj as n → ∞. Since the triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n} of random sets is row-wise exchangeable and by Lemma 3.1(2), it follows that for each j ∈ {1, 2, . . . , m} and for each n ≥ 1, there exists a sequence (j) (j) (j) 1 (FXni ) such that {gni : 1 ≤ i ≤ n} is exchangeable. Since then, we {gni : 1 ≤ i ≤ n} of gni ∈ SX ni (j) (j) (j) get Egni = Egn1 for all i ∈ {1, 2, . . . , n}. It follows that Egni → xj as n → ∞ for each i and j. 1 (FXni ) as follows: Next, we define the triangular array {fni : n ≥ 1, 1 ≤ i ≤ n} of fni ∈ SX ni (j) fni (ω) := gni (ω) if i ≡ j (mod m), where j ∈ {1, 2, . . . , m} and for all ω ∈ Ω. Let yni = Efni and gni (ω) = fni (ω) − yni , where n ≥ 1, 1 ≤ i ≤ n, ω ∈ Ω. Let n = (k − 1)m + l, 1 ≤ l ≤ m. By the arguments as in the proof of Theorem 3.7, we get that the array {gn,(i−1)m+j : n ≥ 1, 1 ≤ i ≤ k} of random elements is row-wise exchangeable, for each j ∈ {1, 2, . . . , m}. By the arguments as in (3.9), for every f ∈ X∗ , E (f (gni )f (gnj )) = Cov (f (fni ), f (fnj )) = Cov (f (g(coXni )), f (g(coXnj ))) = Cov (f (gn (coXn1 )), f (gn (coXn2 ))) → 0 as n → ∞ for all i = j and i ≡ j(mod m). (by the condition (3.30)) (3.34) Similarly, by the condition (3.31), we have that E f 2 (gn1 ) = V ar(f (gn (coXn1 ))) = o(n), for all f ∈ X∗ . As in the proof of Theorem 3.7, for each n = (k − 1)m + l and for each j ∈ {1, 2, . . . , m}, set (j) (j) k Sn (ω) = k1 i=1 gn,(i−1)m+j (ω) for all ω ∈ Ω. For each j ∈ {1, 2, . . . , m}, the sequence {Sn : n ≥ 1} (j) of random elements is divided into m subsequences {S(k−1)m+l : k ≥ 1}, l ∈ {1, 2, . . . , m}. 14 (j) Since the triangular array {gni : n ≥ 1, 1 ≤ i ≤ n} of random elements is row-wise exchangeable (j) and by the condition (3.29), the sequence {E(gnj |G(n, m, j)) : n ≥ 1} of random elements is a reverse martingale, for each m ≥ 1 and j ∈ {1, 2, . . . , m}. For each m ≥ 1 and l, j ∈ {1, 2, . . . , m}, set k (l,j) Gk (f ) k+1 = σ{ fkm+l,(i−1)m+j , . . . }. f(k−1)m+l,(i−1)m+j , i=1 i=1 (j) For each m ≥ 1 and l, j ∈ {1, 2, . . . , m}, the sequence {E(g(k−1)m+l,j |G((k − 1)m + l, m, j)) : k ≥ 1} of random elements is a reverse martingale, since every subsequence of a reverse martingale sequence is also a reverse martingale. (j) (l,j) Next, we will show that the sequence {E(g(k−1)m+l,j |Gk (f )) : k ≥ 1} of random elements is a reverse martingale. Indeed, it suffices to show that (j) (l,j) E(E(g(k−1)m+l,j |Gk (l,j) (j) (l,j) (f ))|Gk+1 (f )) = E(gkm+l,j |Gk+1 (f )) a.s., which is equivalent to (j) (l,j) (j) (l,j) E(g(k−1)m+l,j |Gk+1 (f )) =E(gkm+l,j |Gk+1 (f )) a.s. (l,j) (l,j) (by the smoothing lemma with Gk+1 (f ) ⊂ Gk (f )). (3.35) (j) Since the sequence {E(g(k−1)m+l,j |G((k − 1)m + l, m, j)) : k ≥ 1} of random elements is a reverse martingale and by the similar argument, we obtain (j) (j) E(g(k−1)m+l,j |G(km + l, m, j)) = E(gkm+l,j |G(km + l, m, j)) a.s. (3.36) We have that (j) (l,j) (j) (l,j) E(gkm+l,j |Gk+1 (f )) = E(E(gkm+l,j |G(km + l, m, j))|Gk+1 (f )) a.s. (l,j) (by the smoothing lemma with Gk+1 (f ) ⊂ G(km + l, m, j)) (j) (l,j) = E(E(g(k−1)m+l,j |G(km + l, m, j))|Gk+1 (f )) a.s. (by (3.36)) (j) (l,j) = E(g(k−1)m+l,j |Gk+1 (f )) (l,j) (by the smoothing lemma with Gk+1 (f ) ⊂ G(km + l, m, j)). Thus, (3.35) is proved. Exchangeability of the sequence {f(k−1)m+l,(i−1)m+j : 1 ≤ i ≤ k} implies that the sequence (l,j) {E(g(k−1)m+l,j |Gk (g)) : k ≥ 1} of random elements is a reverse martingale (where k (l,j) Gk (g) = σ{ k+1 gkm+l,(i−1)m+j , . . . }). g(k−1)m+l,(i−1)m+j , i=1 i=1 (j) (j) By (3.28) and by the exchangeability of {gni : 1 ≤ i ≤ n}, we deduce that the sequence {gnj : n ≥ 1} is CUI, for each j ∈ {1, 2, . . . , m}. This yields that {f(k−1)m+l,j : k ≥ 1} is CUI, for each l, j ∈ (j) {1, 2, . . . , m} (Because g(k−1)m+l,j (ω) = f(k−1)m+l,j (ω)). Moreover, the sequence {Ef(k−1)m+l,j : k ≥ 1} converges to xj as k → ∞. Applying Lemma 3.10, we obtain that the sequence {g(k−1)m+l,j : k ≥ 1} is CUI, for each l, j ∈ {1, 2, . . . , m}. Hence, for each l, j ∈ {1, 2, . . . , m}, the triangular array {g(k−1)m+l,(i−1)m+j : k ≥ 1, 1 ≤ i ≤ k} satisfies all the conditions of Lemma 3.3, and so (j) S(k−1)m+l (ω) 1 = k k g(k−1)m+l,(i−1)m+j (ω) → 0 a.s. as k → ∞, i=1 15 for each l, j ∈ {1, 2, . . . , m}. It is equivalent to Sn(j) (ω) = (j) 1 k k gn,(i−1)m+j (ω) → 0 a.s. as n → ∞, for each j ∈ {1, 2, . . . , m}. i=1 (l,j) Suppose that Vn , zki are defined as in the proof of Theorem 3.7. For each j ∈ {1, 2, . . . , m}, since the array {fn,(i−1)m+j : n ≥ 1, 1 ≤ i ≤ k} is row-wise exchangeable and {Efn,(i−1)m+j } converges to xj as n → ∞ for each column, it implies that this array has bounded expectation. Therefore, we have (l,j) |zki | ≤ Efn,(i−1)m+j + xj ≤ C + xj , for all k ≥ 1, 1 ≤ i ≤ k, n = (k − 1)m + l. (l,j) (l,j) By zki → 0 as k → ∞, for each i ≥ 1, we get zki → 0 as i → ∞. Combining the above statements, we have that for each m ≥ 1 and l, j ∈ {1, 2, . . . , m}, the (l,j) triangular array {zki : k ≥ 1, 1 ≤ i ≤ k} satisfies all the conditions of Lemma 3.5. To complete the “lim inf” path of the proof in the Mosco convergence, we proceed as in the proof of Theorem 3.7. Finally, by the arguments as in the proof of Theorem 3.7 and by Lemma 3.4, we obtain the “lim sup” path of the Mosco convergence. Remark. 1. In Theorem 3.11, if the condition (3.29) is replaced by the following two conditions: +) {E(f (g(coXn1 ))|Gg (n, m, j)) : n ≥ 1} is a reverse martingale, for each m ≥ 1, 1 ≤ j ≤ m, f ∈ X∗ , g +) {E(g(coXn1 )I[g(coXn1 )∈K] / |G (n, m, j)) : n ≥ 1} is a reverse martingale, for each m ≥ 1, 1 ≤ j ≤ m, and for each compact subset K of X, then by the same arguments as in the proof of Theorem 3.11 and using [18, Theorem 3.3], the SLLN also holds. 2. In the past results, one built the family of selections of random sets to prove the “lim inf” path by being the union of the families with respect to xj , j ∈ {1, 2, . . . , m}. However, in present paper, the triangular array {fni : n ≥ 1, 1 ≤ i ≤ n} of selections of random sets is the union of sets which each set is a sub family of triangular array with respect to xj , j ∈ {1, 2, . . . , m}. Then, we use the single-valued SLLN for each triangular sub-array to obtain the multivalued SLLN. This is one of the key tools to prove the most difficult “half” of the multivalued SLLN in the Mosco topology. The example below shows that Theorem 3.11 is really different from Lemma 3.3, even in the case of single-valued random variables. Example 1. Consider the Banach space X = R. The triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n} is defined by Xni (ω) = {1} for every n ≥ 1, 1 ≤ i ≤ n and ω ∈ Ω. Then, it is easy to check that the triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n} satisfies all the conditions of Theorem 3.11. But, it follows 2 = 1 for all n ≥ 1 that the triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n} does not satisfy the from EXn1 condition (iii) of Lemma 3.3. However, the conditions (3.29) and (3.32) in Theorem 3.11 are also necessary. The next example shows that Theorem 3.11 is not true without the conditions (3.29) and (3.32). Example 2. Let X = 2 be the space of square-summable sequences. Namely, x ∈ 2 if x = ∞ ∞ 2 (x1 , x2 , ..., xn , ...), xi ∈ R and n=1 |xn |2 < ∞. The norm · l2 is defined by x l2 = n=1 |xn | . ∞ 2 Then, is a Hilbert space with scalar multiplication (·|·) which is given by (x|y) = n=1 xn yn for each x = (x1 , x2 , ..., xn , ...) ∈ 2 , y = (y1 , y2 , ..., yn , ...) ∈ 2 . For each i ≥ 1, let ei = {0, ..., 0, 1, 0, ...}, with number 1 in the ith position. Then, {e1 , e2 , ..., en , ...} is a standard basis of X. 16 For each n ≥ 1, 1 ≤ i ≤ n and ω ∈ Ω, we set Xni (ω) = {en }. Then, the triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n} satisfies all the conditions of Theorem 3.11 without the conditions (3.29) and (3.32). We √ n have Gn (ω) = n1 cl i=1 Xni (ω) = {en } for every n ≥ 1 and ω ∈ Ω. Since em − en l2 = 2 for all m = n, the sequence {en : n ≥ 1} is not Cauchy’s sequence. Consequently, the sequence {en : n ≥ 1} does not converge in norm. Therefore, we have 0 ∈ / s-liGn (ω) for all ω ∈ Ω. By Riezs’s theorem, we have that for each f ∈ X∗ , there exists a ∈ 2 such that f (x) = (a|x) for ∞ all x ∈ X. On the other hand, a = n=1 (a|en ).en ∈ 2 . This series converges to a. It implies that the general term (a|en ) converges to 0 as n → ∞, which is equivalent to lim f (en ) = f (0). It follows that w en → 0 as n → ∞. Since then, 0 ∈ w-lsGn (ω) for all ω ∈ Ω. Since the above statements, we do not obtain the SLLN for the triangular array {Xni : n ≥ 1, 1 ≤ i ≤ n} with respect to Mosco convergence. In Theorems 3.7 and 3.11, we use a condition which is general stronger than the condition (i) of Lemma 3.5, that is, (l,j) zki → 0 as k → ∞ for each i. (iii) However, the condition (ii) is also necessary in this case. Indeed, the following example shows that if the condition (i) is replaced by the condition (iii) then Lemma 3.5 without condition (ii) is also not true. Example 3. Let {xni : n ≥ 1, 1 ≤ i ≤ n} be a triangular array of elements in R and it is defined as follows: n2 if i = n, xni = 1 otherwise. n This means that xni n≥1,1≤i≤n 12 1 2 1 = 3 .. . 1 n .. . 22 1 3 .. . 32 .. . 1 n 1 n .. . .. . .. . ... ... n2 .. . .. . It is clear that limn→∞ xni = 0 for each i; namely, the condition (iii) is satisfied. Moreover, xnn = n2 → ∞ as n → ∞. This means that the condition (ii) of Lemma 3.5 is not satisfied. However, n 1 1 n−1 n−1 xni = ( + n2 ) = + n → ∞ as n → ∞. n i=1 n n n2 Now, we extend the previous theorems to fuzzy-valued random sets. ˜ ni : n ≥ 1, 1 ≤ i ≤ n} be a triangular array of row-wise exchangeable fuzzy Theorem 3.12. Let {X ˜ ni : n ≥ 1, 1 ≤ i ≤ random sets such that for each α ∈ (0, 1], the triangular array of random sets {Lα X n} satisfies all the conditions of one of three Theorems 3.7, 3.9 and 3.11. Then, n M - lim 1 ˜ ni (ω) = IcoX a.s., cl X n i=1 where IcoX is the indicator function of coX. ˜ n (ω) = 1 cl n X ˜ Proof. Let G i=1 ni (ω). By virtue of the suitable theorem (one of three Theorems 3.7, n ˜ n (ω) = coX a.s. for every fixed α ∈ (0, 1], in particular, for 3.9 and 3.11), we have that M - lim Lα G 17 every α = r ∈ Q, where Q is the set of all rational numbers. Since countable set Q is dense in [0, 1] ˜ n (ω) = limr↑α,r∈Q Lr G ˜ n (ω), we have M - lim Lα G ˜ n (ω) = coX, for every α ∈ (0, 1], a.s. and Lα G Next, for each C ∈ c(X), there exists an unit (with probability one) fuzzy-valued random set Y˜ satisfying Lα Y˜ (ω) = C, for all α ∈ (0, 1], a.s. Indeed, it is easy to check that Lα IC = C, for all α ∈ (0, 1]. Suppose that the fuzzy random set Y˜ satisfying Lα Y˜ (ω) = C for all α ∈ (0, 1] a.s. For each ω ∈ N with P(N ) = 1, put u = Y˜ (ω). It follows from the sets Lα u, α ∈ (0, 1] are non-increasing monotonic ordered by inclusion as α ↑ that Lα u = C for all α ∈ (0, 1] is equivalent to L0+ u ⊂ C ⊂ L1 u, where L0+ u = {x ∈ X | u(x) > 0}. Since then, it is not hard to prove that u = IC , which implies Y˜ (ω) = IC a.s. ˜ n (ω) = IcoX a.s. ˜ n (ω) = Lα IcoX for every α ∈ (0, 1], a.s., that is, M - lim G Hence, M - lim Lα G Acknowledgments The paper was done when the authors were visit to Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank the VIASM for their kind hospitality and support. References [1] Artstein, Z. and Vitale, R. A. (1975). A strong law of large numbers for random compact sets. The Annals of Probability, 3, 5, 879-882. [2] Castaing, C., Quang, N. V. and Giap, D. X. (2012). Mosco convergence of strong law of large numbers for double array of closed valued random variables in Banach space. Journal of Nonlinear and Convex Analysis, 13, 4, 615-636. [3] Castaing, C., Quang, N. V. and Giap, D. X. (2012). Various convergence results in strong law of large numbers for double array of random sets in Banach spaces. Journal of Nonlinear and Convex Analysis, 13, 1, 1-30. 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Stochastic Analysis and Application, 15, 1, 51-57. [10] Li, S. and Ogura, Y. (1998). Convergence of set valued sub- and supermartingales in the Kuratowski-Mosco sense. The Annals of Probability, 26, 3, 1384-1402. [11] Li, S. and Ogura, Y. (1999). Convergence of set-valued and fuzzy-valued martingales. Fuzzy Sets and Systems, 101, 3, 453-461. 18 [12] Inoue, H. (1995). Randomly weighted sums for exchangeable fuzzy random variables. Fuzzy Sets and Systems, 69, 3, 347-354. [13] Inoue, H. (2001). Exchangeability and convergence for random sets. Information sciences, 133, 1-2, 23-37. [14] Inoue, H. and Taylor, R. L. (2006). Law of large numbers for exchangeable random sets in Kuratowski-Mosco sense. Stochastic Analysis and Applications, 24, 2, pp. 263-275. [15] Kim, Y. K. (2000). A strong law of large numbers for fuzzy random variables. Fuzzy Sets and Systems, 111, 3, 319-323. [16] Mosco, U. (1969). Convergence of convex sets and of solutions of variational inequalities. Advances in Mathematics, 3, 4, 510-585. [17] Mosco, U. (1971). On the continuity of the Young-Fenchel transform. Journal of Mathematical Analysis and Applications, 35, 518-535. [18] Patterson, R. F. and Taylor, R. L. (1985). Strong laws of large numbers for triangular arrays of exchangeable random variables. Stochastic Analysis and Applications, 3, 2, 171-187. [19] Puri, M. L. and Ralescu, D. A. (1983). Strong law of large numbers for Banach space valued random sets. The Annals of Probability, 11, 1, 222-224. [20] Puri, M. L. and Ralescu, D. A. (1986). Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114, 2, 409-422. [21] Quang, N. V. and Giap, D. X. (2013). Mosco convergence of SLLN for triangular arrays of rowwise independent random sets. Statistics and Probability Letters, 83, 4, 1117-1126. [22] Quang, N. V. and Giap, D. X. (2013). SLLN for double array of mixing dependence random sets and fuzzy random sets in a separable Banach space. Journal of Convex Analysis, 20, 4. [23] Quang, N. V. and Thuan, N. T. (2012). Strong laws of large numbers for adapted arrays of setvalued and fuzzy-valued random variables in Banach space. Fuzzy Sets and Systems, 209, 14-32. [24] Taylor, R. L. and Patterson, R. F. (1985). Strong laws of large numbers for arrays of row-wise exchangeable random elements. International Journal of Mathematics and Mathematical Sciences, 8, 1, 135-144. 19 [...]... C., Quang, N V and Giap, D X (2012) Various convergence results in strong law of large numbers for double array of random sets in Banach spaces Journal of Nonlinear and Convex Analysis, 13, 1, 1-30 [4] Fu, K A and Zhang, L X (2008) Strong laws of large numbers for arrays of rowwise independent random compact sets and fuzzy random sets Fuzzy Sets and Systems, 159, 24, 3360-3368 [5] Fu, K A and Zhang, L... and convergence for random sets Information sciences, 133, 1-2, 23-37 [14] Inoue, H and Taylor, R L (2006) Law of large numbers for exchangeable random sets in Kuratowski -Mosco sense Stochastic Analysis and Applications, 24, 2, pp 263-275 [15] Kim, Y K (2000) A strong law of large numbers for fuzzy random variables Fuzzy Sets and Systems, 111, 3, 319-323 [16] Mosco, U (1969) Convergence of convex sets. .. union of the families with respect to xj , j ∈ {1, 2, , m} However, in present paper, the triangular array {fni : n ≥ 1, 1 ≤ i ≤ n} of selections of random sets is the union of sets which each set is a sub family of triangular array with respect to xj , j ∈ {1, 2, , m} Then, we use the single-valued SLLN for each triangular sub -array to obtain the multivalued SLLN This is one of the key tools to. .. Puri, M L and Ralescu, D A (1983) Strong law of large numbers for Banach space valued random sets The Annals of Probability, 11, 1, 222-224 [20] Puri, M L and Ralescu, D A (1986) Fuzzy random variables Journal of Mathematical Analysis and Applications, 114, 2, 409-422 [21] Quang, N V and Giap, D X (2013) Mosco convergence of SLLN for triangular arrays of rowwise independent random sets Statistics and Probability... V and Giap, D X (2013) SLLN for double array of mixing dependence random sets and fuzzy random sets in a separable Banach space Journal of Convex Analysis, 20, 4 [23] Quang, N V and Thuan, N T (2012) Strong laws of large numbers for adapted arrays of setvalued and fuzzy- valued random variables in Banach space Fuzzy Sets and Systems, 209, 14-32 [24] Taylor, R L and Patterson, R F (1985) Strong laws of. .. theorems to fuzzy- valued random sets ˜ ni : n ≥ 1, 1 ≤ i ≤ n} be a triangular array of row- wise exchangeable fuzzy Theorem 3.12 Let {X ˜ ni : n ≥ 1, 1 ≤ i ≤ random sets such that for each α ∈ (0, 1], the triangular array of random sets {Lα X n} satisfies all the conditions of one of three Theorems 3.7, 3.9 and 3.11 Then, n M - lim 1 ˜ ni (ω) = IcoX a.s., cl X n i=1 where IcoX is the indicator function of. .. and Ogura, Y (1998) Convergence of set valued sub- and supermartingales in the Kuratowski -Mosco sense The Annals of Probability, 26, 3, 1384-1402 [11] Li, S and Ogura, Y (1999) Convergence of set-valued and fuzzy- valued martingales Fuzzy Sets and Systems, 101, 3, 453-461 18 [12] Inoue, H (1995) Randomly weighted sums for exchangeable fuzzy random variables Fuzzy Sets and Systems, 69, 3, 347-354 [13]... like to thank the VIASM for their kind hospitality and support References [1] Artstein, Z and Vitale, R A (1975) A strong law of large numbers for random compact sets The Annals of Probability, 3, 5, 879-882 [2] Castaing, C., Quang, N V and Giap, D X (2012) Mosco convergence of strong law of large numbers for double array of closed valued random variables in Banach space Journal of Nonlinear and Convex... 3.9, we obtain the SLLN as in Lemma 3.2 Next, we will establish a multivalued SLLN for triangular array of row- wise exchangeable random sets with CUI and reverse martingale conditions To do this, we need the following lemma Lemma 3.10 Let {fn : n ≥ 1} be a sequence of random elements in L1 (X) Suppose that the sequence {fn : n ≥ 1} is CUI and Efn → x as n → ∞, where x is an element of X Then, the sequence... n → ∞, (3.21) for all s = k, f ∈ X∗ , for each j ∈ {1, 2, , m} By the arguments as in the proof of Theorem 3.7, we have that for each j ∈ {1, 2, , m}, the rowwise exchangeability of array {fn,(i−1)m+j } of random elements implies the row- wise exchangeability of array {fn,(i−1)m+j − xj } (j) k For each n = (k − 1)m + l, we put Sn (ω) = k1 i=1 (fn,(i−1)m+j (ω) − xj ) for all ω ∈ Ω For each (j) j ... concept of i.i.d random sets SLLN in Mosco convergence for triangular array of rowwise exchangeable random sets (F) (resp SY1 (F)) If X, Y are Let X, Y be two random sets and f (resp g) belongs to. .. of rowwise independent random compact sets and fuzzy random sets Fuzzy Sets and Systems, 159, 24, 3360-3368 [5] Fu, K A and Zhang, L X (2008) Strong limit theorems for random sets and fuzzy random. .. setvalued random variable, fuzzy- valued random variable, Mosco convergence and exchangeability Section is concerned with some theorems on Mosco convergence of the SLLN for triangular arrays of row- wise