A Strong Law of Large Numbers for Set-Valued Random Variables in Gα Space

In this paper, we shall represent a strong law of large numbers (SLLN) for weighted sums of set-valued random variables in the sense of the Hausdorff metric dH , based on the result of single-valued random variable obtained by Taylor [11].


Introduction
We all know that the limit theories are important in probability and statistics. For singlevalued case, many beautiful results for limit theory have been obtained. In [11], there are many results of laws of large numbers at different kinds of conditions and different kinds of spaces. With the development of set-valued random theory, the theory of set-valued random variables and their applications have become one of new and active branches in probability theory. And the theory of set-valued random variables has been developed quite extensively (cf. [1], [2], [6], [7], [8], [10] etc.). In [1], Artstein and Vitale used an embedding theorem to prove a strong law of large numbers for independent and identically distributed set-valued random variables whose basic space is R d , and Hiai extended it to separable Banach space X in [5]. Taylor and Inoue proved SLLN's for only independent case in Banach space in [10]. Many other authors such as Giné, Hahn and Zinn [4], Puri and Ralescu [9] discussed SLLN's under different settings for set-valued random variables where the underlying space is a separable Banach space.
In this paper, what we concerned is the SLLN of set-valued independent random variables in G α space. Here the geometric conditions are imposed on the Banach spaces to obtain SLLN for set-valued random variables. The results are both the extension of the single-valued's case and the extension of the set-valued's case.
This paper is organized as follows. In section 2, we shall briefly introduce some definitions and basic results of set-valued random variables. In section 3, we shall prove a strong law of large numbers for set-valued independent random variables in G α space.

Preliminaries on set-valued random variables
Throughout this paper, we assume that (Ω, A, µ) is a nonatomic complete probability space, (X, ∥ · ∥) is a real separable Banach space, N is the set of nature numbers, K(X) is the family of all nonempty closed subsets of X, and K bc (X) is the family of all nonempty bounded closed 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 as The Hausdorff metric on K(X) is defined by For more general hyperspaces, more topological properties of hyperspaces, readers may refer to a good book [3]. 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 * The following is the equivalent definition of Hausdorff metric. For each A, B ∈ K bc (X), For each set-valued random variable F , the expectation of F , denoted by E[F ], is defined as where ∫ Ω f dµ is the usual Bochner integral in L 1 [Ω, X], the family of integrable X-valued random variables, and

Main Results
In this section , we will give the limit theorems for independent set-valued random variables in G α space. The following definition and lemma are from [11], which will be used later.
Definition 3.1 A Banach space X is said to satisfy the condition G α for some α, 0 < α ≤ 1, if there exists a mapping G : X → X * such that for all x, y ∈ X and some positive constant A.
Note that Hilbert spaces are G 1 with constant A = 1 and identity mapping G. Lemma 3.2 Let X be a separable Banach space which is G α for some 0 < α ≤ 1 and let {V 1 , V 2 , · · · , V n } be single-valued independent random elements in X such that E[V k ] = 0 and E[∥V k ∥ 1+α ] < ∞ for each k = 1, 2, · · · , n. then where A is the positive constant in (iii). Theorem 3.2 Let X be a separable Banach space which is G α for some 0 < α ≤ 1. Let {F n : n ≥ 1} be a sequence of independent set-valued random variables in K bc (X), such that Note that F j = U j + W j for each j and that both {U j : j ≥ 1} and {W j : j ≥ 1} are independent sequences of set-valued random variables. Next, for each m and n .
That means

is a Cauchy sequence and hence
converges as m −→ ∞. Since convergence in the mean implied convergence in probability, Ito and Nisio's(1968) result for independent random elements(rf. Section 4.5) provides that W j ∥ K converges in probability 1 as n → ∞.
Then for n, m ≥ 1, m > n, by triangular inequality we have .
For any fixed n, m, there exists a sequence x * k ∈ S * , such that Then by dominated convergence theorem, Minkowski inequality and Lemma ??, we have