- Research Article
- Open Access
A Note on Strong Laws of Large Numbers for Dependent Random Sets and Fuzzy Random Sets
© Ke-Ang Fu 2010
- Received: 16 August 2009
- Accepted: 7 February 2010
- Published: 23 March 2010
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.
- Limit Theorem
- Borel Subset
- Separable Banach Space
- Fuzzy Random Variable
- Closed Convex Hull
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  and Matheron  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 , Puri and Ralescu , Hiai , Inoue , Taylor et al. [9–11], Uemura , and so forth. Among them, Artstein and Vitale  proved limit theorems concerning random sets in and Puri and Ralescu  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 .
The theory of fuzzy sets was introduced by Zadeh  (for an outline recently, one can refer to [14, 15]), and the concept of fuzzy random variables was promoted by Kwakernaak , where useful basic properties were developed. Puri and Ralescu  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  proved an SLLN for i.i.d. fuzzy random variables. Klement et al.  considered fuzzy versions of random sets in Euclidean spaces and obtained an i.i.d. SLLN. Inoue  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 [8–11, 21–26]. Also for more detailed results about limit theorems of random sets and fuzzy random sets, we refer the readers to Li et al.  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. , and Terán . 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.
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. ).
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 . If a Bochner integral can be defined as and (cf. ). The random set is said to be integrably bounded if the real-valued random variable is integrable (cf. [27, 32]). Hiai and Umegaki  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,
Furthermore, for any
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.
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.
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.
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.
Noting that is measurable, it follows that
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.
The lemma also holds for -mixing random closed sets in a similar way.
Hiai  proved a strong law of large numbers of i.i.d. random variables in in Kuratowski-Mosco convergence. Recently, Inoue and Taylor  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.
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
which implies Thus a.s. follows.
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
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.
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.
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).
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