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ON MARCINKIEWICZ'S TYPE LAW FOR FUZZY RANDOM SETS
ON MARCINKIEWICZ'S TYPE LAW FOR FUZZY RANDOM SETS
Journal of Applied Mathematics & Informatics. 2014. Jan, 32(1_2): 55-60
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : July 03, 2013
  • Accepted : September 12, 2013
  • Published : January 28, 2014
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About the Authors
JOONG-SUNG KWON
HONG-TAE SHIM

Abstract
In this paper, we will obtain Marcinkiewicz's type limit laws for fuzzy random sets as follows : Let { Xn | n ≥ 1} be a sequence of independent identically distributed fuzzy random sets and with 1≤ r ≤ 2. Then the following are equivalent: a.s. in the metric ρp if and only if in probability in the metric ρp if and only if in L 1 if and only if in L r where . AMS Mathematics Subject Classification : 60F05. 60G05.
Keywords
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 ) ( K c ( 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
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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
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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 A α the α -level set of A (that is Aα = { x Rn : A ( x ) ≥ α } ) for all α ∈ (0, 1] and by A 0 the closure of the support of A (that is A 0 = cl { x Rn : A ( x ) > 0} ).
Let Fc ( Rn )( Fcoc ( Rn )) be the class of the fuzzy sets A satisfying the following conditions
  • (1)A1≠ ∅,
  • (2)A0is compact, and
  • (3)Ais upper semi continuous
  • ((4)Aαis convex for allα∈ [0, 1]).
And
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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
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for all ω ∈ Ω, is a random variable, where
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is the fuzzy set where
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and
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otherwise.
A support function of a non-void bounded subset K of Rn is defined by
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where < x , y > denotes the standard scalar product of the vectors x and y . Support functions sK are uniquely associated with the subsets K K c ( Rn ) and preserve addition and nonnegative scalar multiplication when we restricted ourselves to K ( Rn ), i.e.
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Now we endow Fc ( Rn ) with the initial topology generated by the mappings
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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
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and
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where
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denotes the unit Lebesgue measure on the unit sphere in Rn . Then dp ( ρp ) becomes a separable metric on
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with the relation ρp dp which induce the same topology
Now consider Lp ([0, 1]× S n−1 ), the Lp -space with respect to [0, 1]× S n−1 , the obvious product σ -algebra and the product measure
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. Then under the Lp -norm ∥∥ p we obtain Lp ([0, 1]× S n−1 ) as a separable Banach space. Next we can embed
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isometrically isomorphic into Lp ([0, 1]× S n−1 ) as a positive cone (for details see [5 , 7] ). Embedding
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into Lp ([0, 1]× S n−1 ), we draw a convergence theorem in Banach space. For 1 ≤ p < ∞, Lp ([0, 1] × S n−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
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and Lp ([0, 1] × S n−1 ) isometrically.
Lemma 3.1. Let 1 ≤ p < ∞ be fixed . Then j :
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Lp ([0, 1] × S n−1 ) by A
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defines an injection mapping satisfying
  • (1)∥j(A) −j(B)∥p=ρp(A,B)
  • (2)j(A+B) =j(A) +j(B)
  • (3)j(λA) = λj(A)
  • for any
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
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for 0 < r < ∞, that is , for any t > 0,
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. Then
  • (i)
  • (ii)
Proof. . Notice that
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is a sequence of random variables stochastically dominated by
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. Now apply Stout’s result.
Lemma 3.3 ( [5] ). Let { Xk |1 ≤ k n } be
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- valued independent random variables . Let
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Then
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Lemma 3.4 ( [5] ). Let { Xk |1 ≤ k n } be independent
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- valued random variables with
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for k = 1, 2, · · · n and 1 ≤ r ≤ 2. Then we have
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where Cr is a positive constant depending only on r ; if r = 2 then it is possible to take C 2 = 4.
Theorem 3.5. Let { Xn | n ≥ 1} be a sequence of independent identically distributed
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- valued fuzzy random variables with
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for 1≤ r ≤2 and let
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. Then the following are equivalent:
  • (i)a.s. in the metric ρp;
  • (ii)in probability in the metric ρp;
  • (iii)in L1
  • (iv)in Lr
Proof . Let j :
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Lp ([0, 1]× S n−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] × S n−1 ). Since
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in what follow we use X and j X interchangeably.
  • Let
First we show that ( i ) ⇔ ( ii ) ⇔ ( iii ) in L1 . Since
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, we have
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Hence it is enough to show that
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Since
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by lemma 1, these equivalence hold by applying Theorem 5 in [5] to
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Now it remains to show that (iii) ⇒ (iv). Assume that
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in L 1 .
Then now
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Thus it is enough to show that
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From Lemma 4
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By a standard calculation, we have
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Thus the proof is completed.
Remark 3.1. (1) For i.i.d real valued random variables, Pyke and Root [13] showed that
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(2) For i.i.d. B-valued random variables with E X 1 ∥ < ∞ for 1 ≤ r < 2, Choi and Sung [2] showed that
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BIO
Joong-Sung Kwon received his Ph.D at University of Washington. Since 1992 he has been a professor at Sunmoon University. His research interest is stochastic limit theory and fuzzy set theory.
Department of Mathematics, Sun Moon University, 100 Kalsan-ri, Tangieong-myeon, Asansi, Choongnam, 336-840, Korea.
e-mail: jskwon@sunmoon.ac.kr
Hong-Tae Shim received Ph. D from the University of Wisconsin-Milwaukee. His research interests center on wavelet theories, Sampling theories and Gibbs' phenomenon for series of special functions.
Department of Mathematics, Sun Moon University, 100 Kalsan-ri, Tangieong-myeon, Asansi, Choongnam, 336-840, Korea.
e-mail: hongtae@sunmoon.ac.kr
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