ON MARCINKIEWICZ'S TYPE LAW FOR FUZZY RANDOM SETS

Abstract

In this paper, we will obtain Marcinkiewicz's type limit laws for fuzzy random sets as follows : Let {$X_n{\mid}n{\geq}1$} be a sequence of independent identically distributed fuzzy random sets and $E{\parallel}X_i{\parallel}^r_{{\rho_p}}$ < ${\infty}$ with $1{\leq}r{\leq}2$. Then the following are equivalent: $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ a.s. in the metric ${\rho}_p$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in probability in the metric ${\rho}_p$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in $L_1$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in $L_r$ where $S_n={\Sigma}^n_{i=1}\;X_i$.

1. Introduction

The study of the fuzzy random sets, defined as measurable mappings on a probability space, was initiated by Kwakernaak [12] where useful basic properties were developed. Puri and Ralescu [9] used the concept of fuzzy random variables in generating results for random sets to fuzzy random sets. Kruse [8] proved a strong law of large numbers for independent identically distributed fuzzy random variables. Artstein and Vitale [1] proved a strong law of large numbers(SLLN) for Rp-valued random sets and Cressie [3] proved a SLLN for some paticular class of Rp-valued random sets. Using Rådstrom embedding(e.g. Rådstrom [14]), Puri and Ralescu [12] proved a SLLN for Banach space valued random sets and they also proved SLLN for fuzzy random sets, which generalized all of previous SLLN for random sets. In recent year, Joo, Kim and Kwon [6] proved Chung’s type law of large numbers for fuzzy random variables and Kwon and Shim [11] obtained a uniform strong law of large numbers for partial sum processes of fuzzy random sets. In this paper we obtain Marcinkiewicz’s type laws for fuzzy random sets in the Euclidean space under the assumption that WLLN holds. The proofs of the results are based heavily on isometrical embeddings of the fuzzy sample spaces, endowed with Lp-metrics, into Lp-spaces. Our results give the fuzzy version of Marcinkiewicz’s type law of large numbers in general Banach spaces.

 

2. Preliminaries

Let K(Rn) (Kc(Rn) ) be the collection of nonempty compact (and convex ) subsets of Euclidean space Rn. The set can be viewed as a linear structure induced by the scalar multiplication and the Minkowski addition, that is

for all A, B ∈ K(Rn) and λ ∈ R. If d is the Hausdoff metric on K(Rn) which, for A, B ∈ K(Rn) , is given by

where |·| denotes the Euclidean norm. Then (K(Rn) , d) is a complete separable metric space [4,10].

A fuzzy set of Rn is a mapping A : Rn → [0, 1]. We will denote by Aα the α-level set of A (that is Aα = {x ∈ Rn : A(x) ≥ α} ) for all α ∈ (0, 1] and by A0 the closure of the support of A(that is A0 = cl{x ∈ Rn : A(x) > 0} ).

Let Fc(Rn)(Fcoc(Rn)) be the class of the fuzzy sets A satisfying the following conditions

And is the subset of Fc(Rn)( Fcoc(Rn)) with bounded support.

Given a measurable space (Ω,A) and the metric space (K(Rn) , d), a random set (or as a random compact set) is associated with a Borel measurable mapping X : Ω → K(Rn) . If X : Ω → K(Rn) is a set-valued mapping, then X is a random set if and only if X−1(C) = {ω ∈ Ω : X(ω) ∩ C ≠ ∅} ∈ A for all C ∈ K(Rn) .

If X is a random set, the mapping denoted by ∥X∥d and defined by

for all ω ∈ Ω, is a random variable, where is the fuzzy set where and otherwise.

A support function of a non-void bounded subset K of Rn is defined by

where denotes the standard scalar product of the vectors x and y. Support functions sK are uniquely associated with the subsets K ∈ Kc(Rn) and preserve addition and nonnegative scalar multiplication when we restricted ourselves to K(Rn), i.e.

Now we endow Fc(Rn) with the initial topology generated by the mappings

then the topology mentioned above enables us to introduce a measurability concept for defining fuzzy random variable. We call a mapping X : Ω → Fc(Rn) fuzzy random variable over (Ω, A, μ) if it is A-measurable over the initial topology.

For a real number p ≥ 1 and A, B ∈ Fc(Rn) , define

and

where denotes the unit Lebesgue measure on the unit sphere in Rn. Then dp(ρp) becomes a separable metric on with the relation ρp ≤ dp which induce the same topology

Now consider Lp([0, 1]×Sn−1), the Lp-space with respect to [0, 1]×Sn−1, the obvious product σ-algebra and the product measure . Then under the Lp-norm ∥∥p we obtain Lp([0, 1]×Sn−1) as a separable Banach space. Next we can embed isometrically isomorphic into Lp([0, 1]×Sn−1) as a positive cone (for details see [5,7]). Embedding into Lp([0, 1]×Sn−1), we draw a convergence theorem in Banach space. For 1 ≤ p < ∞, Lp([0, 1] × Sn−1) is so called separable Banach space of type min(p, 2). It is known that separable Banach spaces of type 2 is exactly those separable Banach space where the classical strong law of large numbers for independent non-identically distributed random variables holds.

 

3. Main Results

To prove the main theorem we will need the following lemmas. Lemma 1 connects two metric spaces and Lp([0, 1] × Sn−1) isometrically.

Lemma 3.1. Let 1 ≤ p < ∞ be fixed. Then j : → Lp([0, 1] × Sn−1) by A defines an injection mapping satisfying

The following is a generalization of a classical result [15, p. 127-128].

Lemma 3.2. Let {Xn|n ≥ 1} be a sequence of fuzzy random sets stochastically dominated by X with for 0 < r < ∞, that is, for any t > 0, . Then

Proof. . Notice that is a sequence of random variables stochastically dominated by . Now apply Stout’s result.

Lemma 3.3 ([5]). Let {Xk|1 ≤ k ≤ n} be -valued independent random variables. Let Then

Lemma 3.4 ([5]). Let {Xk|1 ≤ k ≤ n} be independent -valued random variables with for k = 1, 2, · · · n and 1 ≤ r ≤ 2. Then we have

where Cr is a positive constant depending only on r ; if r = 2 then it is possible to take C2 = 4.

Theorem 3.5. Let {Xn|n ≥ 1} be a sequence of independent identically distributed -valued fuzzy random variables with for 1≤r≤2 and let .Then the following are equivalent:

Proof. Let j : → Lp([0, 1]×Sn−1) be an isometry. Then {j ◦Xn|n ≥ 1} be a sequence of independent identically distributed random element in a Banach space Lp([0, 1] × Sn−1). Since in what follow we use X and j ◦ X interchangeably.

First we show that (i) ⇔ (ii) ⇔ (iii) in L1. Since , we have

Hence it is enough to show that

Since by lemma 1, these equivalence hold by applying Theorem 5 in [5] to

Now it remains to show that (iii) ⇒ (iv). Assume that in L1.

Then now

Thus it is enough to show that

From Lemma 4

By a standard calculation, we have Thus the proof is completed.

Remark 3.1. (1) For i.i.d real valued random variables, Pyke and Root [13] showed that

(2) For i.i.d. B-valued random variables with E∥X1∥ < ∞ for 1 ≤ r < 2, Choi and Sung [2] showed that