Received 30 March 2015; accepted 23 June 2015; published 30 June 2015

ABSTRACT

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 d_H, based on the result of single-va- lued random variable obtained by Taylor [1].

Keywords:

Set-Valued Random Variable, the Laws of Large Numbers, Hausdorff Metric

1. Introduction

We all know that the limit theories are important in probability and statistics. For single-valued case, many beautiful results for limit theory have been obtained. In [1], 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. [2]- [7] 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, and Hiai extended it to separable Banach space in [8]. Taylor and Inoue proved SLLN's for only independent case in Banach space in [7]. Many other authors such as Giné, Hahn and Zinn [9], Puri and Ralescu [10] 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 space. Here the geometric conditions are imposed on the Banach spaces to obtain SLLN for set-valued random varia- bles. 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 inde- pendent random variables in space.

2. Preliminaries on Set-Valued Random Variables

Throughout this paper, we assume that is a nonatomic complete probability space, is a real separable Banach space, is the set of nature numbers, is the family of all nonempty closed subsets of, and is the family of all nonempty bounded closed convex subsets of.

Let and be two nonempty subsets of and let, the set of all real numbers. We define addi- tion and scalar multiplication as

The Hausdorff metric on is defined by

for. For an in, let. The metric space is complete , and is a closed subset of (cf. [6], Theorems 1.1.2 and 1.1.3). For more general hyper-

spaces, more topological properties of hyperspaces, readers may refer to a good book [11].

For each, define the support function by

where is the dual space of.

Let denote the unit sphere of, the all continuous functions of, and the norm is defined as

The following is the equivalent definition of Hausdorff metric.

For each,

A set-valued mapping is called a set-valued random variable (or a random set, or a multifunction) if, for each open subset of,.

For each set-valued random variable, the expectation of, denoted by, is defined as

where is the usual Bochner integral in, the family of integrable -valued random variables, and. This integral was first introduced by Aumann [3], called Aumann integral in literature.

3. Main Results

In this section, we will give the limit theorems for independent set-valued random variables in space. The following definition and lemma are from [1], which will be used later.

Definition 3.1 A Banach space is said to satisfy the condition for some, if there exists a mapping such that

(i);

(ii);

(iii) for all and some positive constant.

Note that Hilbert spaces are with constant and identity mapping.

Lemma 3.1 Let be a separable Banach space which is for some and let be single-valued independent random elements in such that and for each then

where is the positive constant in (iii).

Theorem 3.1 Let be a separable Banach space which is for some. Let be a sequence of independent set-valued random variables in, such that for each. If

where for and for, then converges with probability 1 in the sense of.

Proof. Define

Note that for each and that both and are independent se- quences of set-valued random variables. Next, for each and

That means is a Cauchy sequence and hence

as. Since convergence in the mean implied convergence in probability, Ito and Nisio’s result in [12] for independent random elements(rf. Section 4.5) provides that

Then for, by triangular inequality we have

By the completeness of, we can have converges almost everywhere in the sense of.

Since by equivalent definition of Hausdorff metric, we have

For any fixed, there exists a sequence, such that

Then by dominated convergence theorem, Minkowski inequality and Lemma 3.1, we have

for each and. Thus, is a Cauchy sequence, and hence converges. Hence, by the similar way as above to prove converges with probability one in the sense of. We also can prove that

with probability one in the sense of. The result was proved. W

From theorem 3.1, we can easily obtain the following corollary.

Corollary 3.2 Let be a separable Banach space which is for some. Let be a sequence of independent set-valued random variables in such that for each. If

are continuous and such that and are non-decreasing, then for each the convergence of

implies that

converges with probability one in the sense of.

Proof. Let

If, by the non-decreasing property of, we have

That is

(4.1)

If, by the non-decreasing property of, we have

That is

(4.2)

Then as the similar proof of theorem 3.1, we can prove both and converges with probability one, and the result was obtained. W

Acknowledgements

The research was supported by NSFC(11301015, 11401016, 11171010), BJNS (1132008).

Cite this paper

Guan Li, (2015) A Strong Law of Large Numbers for Set-Valued Random Variables in Gα Space. Journal of Applied Mathematics and Physics,03,797-801. doi: 10.4236/jamp.2015.37097

References