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A Note on Strong Laws of Large Numbers for Dependent Random Sets and Fuzzy Random Sets

Journal of Inequalities and Applications20102010:286862

https://doi.org/10.1155/2010/286862

Received: 16 August 2009

Accepted: 7 February 2010

Published: 23 March 2010

Abstract

This paper deals with a sequence of identically distributed random sets or fuzzy random sets with -mixing dependence in a separable Banach space. The strong laws of large numbers for these two sequences are derived under Kuratowski-Mosco sense.

Keywords

Limit TheoremBorel SubsetSeparable Banach SpaceFuzzy Random VariableClosed Convex Hull

1. Introduction

Recently, great progress has been made towards the theories and applications of random sets and fuzzy random sets in the areas of information science, probability, and statistics. It is well known that Robbins [1, 2] first proposed the concept of random sets and investigated the relationships between random sets and geometric probabilities in his early work. After that, Kendall [3] and Matheron [4] provided a comprehensive mathematical theory of random sets which was greatly influenced by the geometric probability prospective. Their proposed framework exerted a strong influence on the limit theorems developed in the recent decades. Notice that strong laws of large numbers (SLLNs) play an important role in probability limit theorems, and several variants of SLLNs were built by Artstein and Vitale [5], Puri and Ralescu [6], Hiai [7], Inoue [8], Taylor et al. [911], Uemura [12], and so forth. Among them, Artstein and Vitale [5] proved limit theorems concerning random sets in and Puri and Ralescu [6] were the first to obtain the SLLNs for independent identically distributed (i.i.d.) Banach space-valued compact convex random sets. Among others, SLLNs were obtained under more relaxed conditions, and a detailed survey of these results is available in Taylor and Inoue [10].

The theory of fuzzy sets was introduced by Zadeh [13] (for an outline recently, one can refer to [14, 15]), and the concept of fuzzy random variables was promoted by Kwakernaak [16], where useful basic properties were developed. Puri and Ralescu [17] used the concept of fuzzy random variables in generalizing results for random sets to fuzzy random sets. With respect to laws of large numbers, Kruse [18] proved an SLLN for i.i.d. fuzzy random variables. Klement et al. [19] considered fuzzy versions of random sets in Euclidean spaces and obtained an i.i.d. SLLN. Inoue [20] derived SLLNs for independent, tight fuzzy random sets, and i.i.d. fuzzy random sets in a separable Banach space. Recently, SLLNs have been established under various conditions, and one can refer to the following papers [811, 2126]. Also for more detailed results about limit theorems of random sets and fuzzy random sets, we refer the readers to Li et al. [27] and references therein.

However, to the best of our knowledge, many limit theorems, especially the laws of large numbers, were obtained for independent random sets or fuzzy random sets in the past decades, and little is known of dependent random sets or fuzzy random sets except the exchangeable dependence involved in Inoue [8, 28], Taylor et al. [11], and Terán [26]. In this paper, we aim to propose a new kind of dependence for random sets and fuzzy random sets, and then establish several strong laws of large numbers in Kuratowski-Mosco convergence without the restriction of compactness, where random sets take values of closed subsets in separable Banach spaces.

The layout of this paper is as follows. In Section 2, we give some basic definitions and properties, and the new dependence is proposed in Section 3. In the last section we show several SLLNs for a sequence of dependent random sets and fuzzy random sets, and their proofs.

2. Definitions and Preliminaries

Throughout this paper, let be a real separable Banach space with the norm and the dual space . For each and denote the norm-closure and the closed convex hull of , respectively. Let (resp., ) denote the collections of all nonempty closed (resp., nonempty closed convex) subsets of . Define the Minkowski's addition and scalar multiplication, respectively, in (or ) by
(2.1)
where (or ) and is a real number. Note that neither nor are linear spaces even when one-dimensional Euclidean space. For the distance of and the Hausdorff distance of and , the norm of and the support function of are defined, respectively, by
(2.2)
Let be a sequence of closed sets in We write if for some Rather than this Hausdorff convergence, we here use the Kuratowski-Mosco convergence. Let be the set of all such that for some that is,   and let be the set of all such that (i.e., converges weakly to ) for some and some subsequence of It is easily seen that , and if Thus we say converges to in the Kuratowski-Mosco sense if and only if
(2.3)

Clearly, the Hausdorff convergence is generally stronger than Kuratowski-Mosco convergence, since the former implies the latter when is infinite dimensional, and in finite dimensional spaces they coincide with bounded sets (cf. [29]).

Let denote a probability measure space. A random closed set is a Borel measurable function that is,   for each Moreover, we assume that the random closed sets are measurable in the sequel, where means the Borel subsets of For a random set in there exists a corresponding set in which can be used in defining an expected value. A measurable function is called a measurable selection of if for every Denote by
(2.4)
where denotes the space of measurable functions such that if and only if the random variable is integrable. For each random set , the expectation of , denoted by is defined by
(2.5)

where is the usual Bochner integral in Define for . This definition was introduced by Aumann in 1965 as a natural generalization of the integral of real-valued random variables in [30]. If a Bochner integral can be defined as and (cf. [31]). The random set is said to be integrably bounded if the real-valued random variable is integrable (cf. [27, 32]). Hiai and Umegaki [32] showed that a random set is integrably bounded if and only if is bounded in Thus an integrably bounded random set may take unbounded sets.

Now we introduce some notions of fuzzy random sets. A fuzzy set in is a function Let denote the family of the fuzzy subset satisfying the following conditions:

(a) is upper semicontinuous, that is, the -level set of , that is, is a closed subset of for each ,

(b) has compact closure,

(c) .

A linear structure in is defined by the following operations:
(2.6)
where This of course implies and Then we adopt the metric (see [17, 19, 33]) as a generalization of the Hausdorff metric from to where
(2.7)
where The concept of a fuzzy random set as a generation for a random set was extensively studied by Puri and Ralescu [17]. A fuzzy random set is a function such that for each is a random closed set. The expectation of a fuzzy random set , denoted by is an element in such that for each
(2.8)
where the closure is taken in and By virtue of the existence theorem (cf. [27]), we have an equivalent definition as follows:
(2.9)

Furthermore, for any

3. Mixing Dependence

Many statistical results are concerned with independent and identically distributed (i.i.d.) random sets or fuzzy random sets. While it is not always possible to assume that random sets or fuzzy random sets are independent, the sequence can be often dependent. However, for dependent case, it seems that only the exchangeability is involved. In what follows, we propose a new kind of dependence for random sets which is popular with random variables and random elements. Similarly, it can be defined for fuzzy random sets.

Given two -fields in , write
(3.1)
Let be a sequence of random closed sets on Denote where (the set of all nature numbers). For two nonempty disjoint sets define as Let and be the -fields generated by and respectively. Now we define two mixing coefficients for the sequence of For any real number set
(3.2)

If (resp. tends to zero as then we say that the sequence is -mixing (resp., -mixing). Obviously, a -mixing sequence is a -mixing sequence. Also it is well known that many limit results were derived for real-valued mixing random sequences and random fields in the past thirty years (cf. [34, 35] and references therein). Zhang [36, 37] extended them to the Banach space-valued mixing random fields and established some moment inequalities. As far as we know, there is little concerning the dependent random sets or fuzzy random sets except the exchangeability dependence. The main purpose of this paper is to establish limit theorems for mixing dependent random sets or fuzzy random sets which extend the results of independent case.

4. Limit Theorems

Lemma 4.1.

Let be the smallest sub- -filed of to which and are measurable, respectively. Let be a random closed set and measurable. Then one has the following.

For each with
(4.1)

where

Let be a sequence of -mixing and identically distributed random closed sets. For each where denotes the set of all measurable functions in there exists such that is -mixing.

For each -mixing and identically distributed random closed sets with one has
(4.2)

Proof.

Note that for the conditional expectation of with respect to is given as a function such that
(4.3)
If with then from Hiai and Umegaki [32] it follows that there exists a -measurable satisfying that
(4.4)

Noting that is measurable, it follows that

(4.5)
Recall that and , and hence
(4.6)
Since is separable and is -measurable, there exists a -measurable function satisfying that for every where means the Borel subsets of Now define Note that is an identically distributed sequence of -mixing random sets, and this leads to that is also of mixing dependence, since the definitions of mixing dependence rely on -fields. In fact, the mixing coefficients of are less than those of Thus we have
(4.7)

Noting that the function of into is -measurable, thus is -mixing and identically distributed. Hence, a.s. implies a.s., which leads to

It follows from (2) easily.

Remark 4.2.

The lemma also holds for -mixing random closed sets in a similar way.

Hiai [7] proved a strong law of large numbers of i.i.d. random variables in in Kuratowski-Mosco convergence. Recently, Inoue and Taylor [28] replaced i.i.d. by exchangeability and obtain a strong law of large numbers. Here we replace the i.i.d. by -mixing dependence which is a more extensive dependence and derive strong laws of large numbers for random sets and fuzzy random sets, respectively.

Theorem 4.3.

Let be a sequence of mixing and identically distributed random closed sets in with Suppose that one of the following conditions is satisfied:
(4.8)
Then one has
(4.9)

Proof.

Here we only consider the -mixing case, since the -mixing case can be proved similarly. By Lemma 4.1(2), for a sequence of -mixing random set in , there exists a -measurable function and the corresponding random elements such that for all

Let and Since is -mixing and identically distributed random sets, it follows that is -mixing and identically distributed. For any and , we can choose such that
(4.10)
where is the smallest sub -field of with respect to which is measurable and By Lemma 4.1, there exists a sequence of such that is -mixing and identically distributed for each Let If where then we have
(4.11)

by a similar way of Theorem of Hiai [7] and Theorem of Zhang [37]. Note that are closed sets in which implies a.s. Thus, it follows that a.s., and hence a.s.

In the following we show Let be a sequence dense in Since is separable, by the separation theorem there exists a sequence in with such that
(4.12)
Thus it follows if and only if for all Notice for each , is a sequence of -mixing random variables in since is -mixing, and hence there exists a -null set such that for every and ,
(4.13)
If for then for some So, for each we have
(4.14)

which implies Thus a.s. follows.

Remark 4.4.

If is a finite dimensional Banach space and are compact sets, then Theorem 4.3 still holds in the Hausdorff convergence.

The next theorem describes a strong law of large numbers for mixing fuzzy random sets in

Theorem 4.5.

Let be a sequence of mixing fuzzy random sets taking values in . If , and one of the following conditions is satisfied:
(4.15)
then
(4.16)

Proof.

Since is a sequence of mixing fuzzy random sets in , by the definitions of mixing we have that is mixing random closed sets with for any Thus the desired result follows from Theorem 4.3 immediately.

Remark 4.6.

By the definitions of mixing dependence and -fields, it follows that the mixing coefficients in Theorem 4.5 is less than those in Theorem 4.3.

Declarations

Acknowledgments

The author thanks the referees for pointing out some errors in a previous version, as well as for several comments that have led to improvements in this work. This project was supported by the National Natural Science Foundation of China (nos. 10671176, 10771192, and 10901138) and the Research Grant of Zhejiang Gongshang University (X10-26).

Authors’ Affiliations

(1)
Department of Mathematics, Zhejiang Gongshang University, Hangzhou, China

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Copyright

© Ke-Ang Fu 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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