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Weak laws of large numbers for weighted sums of Banach space valued fuzzy random variables
Weak laws of large numbers for weighted sums of Banach space valued fuzzy random variables
International Journal of Fuzzy Logic and Intelligent Systems. 2013. Sep, 13(3): 215-223
Copyright ©2013, Korean Institute of Intelligent Systems
This is an Open Access article distributedunder the terms of the CreativeCommons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercialuse, distribution, and reproductionin any medium, provided the originalwork is properly cited.
  • Received : August 16, 2013
  • Accepted : September 14, 2013
  • Published : September 25, 2013
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About the Authors
Yun Kyong Kim
ykkim@dsu.ac.kr

Abstract
In this paper, we present some results on weak laws of large numbers for weighted sums of fuzzy random variables taking values in the space of normal and upper-semicontinuous fuzzy sets with compact support in a separable real Banach space. First, we give weak laws of large numbers for weighted sums of strong-compactly uniformly integrable fuzzy random variables. Then, we consider the case that the weighted averages of expectations of fuzzy random variables converge. Finally, weak laws of large numbers for weighted sums of strongly tight or identically distributed fuzzy random variables are obtained as corollaries.
Keywords
1. Introduction
In recent years, the theory of fuzzy sets introduced by Zadeh [1] has been extensively studied and applied the fields of statistics and probability. Statistical inference for fuzzy probability models led to the requirement for laws of large numbers to ensure consistency in estimation problems.
Since Puri and Ralescu [2] introduced the concept of fuzzy random variables as a natural generalization of random sets, several authors have studied laws of large numbers for fuzzy random variables. Among others, several variants of strong law of large numbers (SLLN) for independent fuzzy random variables were built on the basis of SLLN for independent random sets. A rich variety of SLLN for fuzzy random variables can be found in the literature, e.g., Colub et al. [3 , 4] , Feng [5] , Fu and Zhang [6] , Inoue [7] , Klement et al. [8] , Li and Ogura [9] , Molchanov [10] , Proske and Puri [11] .
However, weak laws of large numbers (WLLN) for fuzzy random variables are not as popular as SLLN. Taylor et al. [12] obtained WLLN for fuzzy random variables in a separable Banach space under varying hypotheses of independence, exchangeability, and tightness. Joo [13] established WLLN for convex-compactly uniformly integrable fuzzy random variables taking values in the space of fuzzy numbers in a finite-dimensional Euclidean space.
Generalizing the above results for sums of fuzzy random variables to the case of weighted sums is a significant problem. In this regard, Guan and Li [14] obtained some results on WLLN for weighted sums of fuzzy random variables under a restrictive condition, and Joo et al. [15] established some results on strong convergence for weighted sums of fuzzy random variables different from those of Guan and Li [14] . Moreover, Kim [16] studied WLLN for weighted sums of level-continuous fuzzy random variables.
The purpose of this paper is to present some results on WLLN for the weighted sum of fuzzy random variables taking values in the space of normal and upper-semicontinuous fuzzy sets with compact support in a real separable Banach space. First, we give WLLN for the weighted sum of strong-compactly uniformly integrable fuzzy random variables. Then, we give WLLN for the weighted sum of fuzzy random variables such that the weighted averages of its expectations are convergent.
2. Preliminaries
Let Y be a real separable Banach space with norm |•| and let K ( Y ) denote the family of all non-empty compact subsets of Y . Then the space K ( Y ) is metrizable by the Hausdorff metric h defined by
Lager Image
A norm of A K ( Y ) is defined by
Lager Image
It is well-known that K ( Y ) is complete and separable with respect to the Hausdorff metric h (See Debreu [17] ).
The addition and scalar multiplication on K ( Y ) are defined as usual:
  • A⊕B= {a+b:a∈A,b∈B}, λA= {λa:a∈A}
for A , B K ( Y ) and λ ∈ R .
The convex hull and closed convex hull of A Y are denoted by co ( A ) and
Lager Image
, respectively. If dim ( Y ) < ∞ and A K ( Y ), then co ( A ) ∈ K ( Y ). But if dim ( Y )= ∞, it is well-known that co ( A ) may not be an element of K ( Y ) even though A K ( Y ), but
Lager Image
K ( Y ) if A K ( Y ).
Let F ( Y ) denote the family of all fuzzy sets u : Y → [0,1] with the following properties;
  • (i)uis normal, i.e., there existsx∈Ysuch thatu(x)= 1;
  • (ii)uis upper-semicontinuous;
  • (iii) suppu=cl{x∈Y:u(x) > 0} is compact, wherecl(A) denotes the closure ofAinY.
For a fuzzy subset u of Y , the α-level set of u is defined by
Lager Image
Then it follows immediately that u F ( Y ) if and only if L α u K ( Y ) for each a ∈ [0,1]: If we denote cl { x Y : u ( x ) > α} by L α + u , then
Lager Image
The linear structure on F ( Y ) is also defined as usual;
Lager Image
for u , v F ( Y ) and λ ∈ R , where
Lager Image
denotes the indicator function of {0}.
Then it is known that for each α ∈ [0,1], L α ( u v )= L α u L α v and L α u )= λ L α u .
Recall that a fuzzy subset u of Y is said to be convex if
  • u(λx+(1–λ)y) ≥ min(u(x),u(y)) forx,y∈Yand λ ∈ [0,1].
The convex hull of u is defined by
  • co(u) = inf{v:vis convex andv≥u}.
Then it is known that for each α ∈ [0,1], L α co ( u )= co ( L α u ).
If Y is finite dimensional space and u F ( Y ), then co ( u ) ∈ F ( Y ). But if Y is infinite dimensional space, it may not be true. So we need the notion of the closed convex hull of u . The closed convex hull
Lager Image
of u is defined by
Lager Image
Then it is well-known that
Lager Image
for each α ∈ [0,1] and
Lager Image
The uniform metric d and norm ||•|| on F ( Y ) as usual;
Lager Image
It is well-known that ( F ( Y ), d ) is complete but is not separable (see Klement et al. [8] ).
3. Main Results
Throughout this paper, let (Ω, A , P ) be a probability space. A set-valued function X : Ω → ( K ( Y ), h ) is called a random set if it is measurable. A random set X is said to be integrably bounded if E || X || < ∞. The expectation of integrably bounded random set X is defined by
  • E(X)= {E(ξ) : ξ ∈L(Ω,Y) and ξ(ω) ∈X(ω)a.s.},
where L (Ω, Y ) denotes the class of all Y -valued random variables ξ such that E |ξ| < ∞.
A fuzzy set valued function
Lager Image
is called a fuzzy random variable (or fuzzy random set) if for each
Lager Image
is a random set. It is well-known that if
Lager Image
is measurable, then
Lager Image
is a fuzzy random variable. But the converse is not true (For details, see Colubi et al. [18] , Kim [19] ).
A fuzzy random set
Lager Image
is said to be integrably bounded if
Lager Image
The expectation of integrably bounded fuzzy random variable
Lager Image
is a fuzzy subset
Lager Image
of Y defined by
Lager Image
For more details for expectations of random sets and fuzzy random variables, the readers may refer to Li et al. [20] .
Let
Lager Image
be a sequence of integrably bounded fuzzy random variables and {λ ni } be a double array of real numbers that not necessarily Toeplitz but satisfying
Lager Image
where C > 0 is a constant not depending on n .
The problem that we will consider is to establish sufficient conditions for
Lager Image
where
Lager Image
denotes the closed convex hull of
Lager Image
To this end, we need the concepts of tightness and compact uniform integrability for a sequence of fuzzy random variables.
Definition 3.1. Let { Xn } be a sequence of random sets.
  • (i) {Xn} is said to be tight if for each ε > 0, there exists a compact subsetKof (K(Y),h) such that
  • P(Xn∉K) < ε for alln.
  • (ii) {Xn} is said to be compactly uniformly integrable(CUI) if for each ε > 0, there exists a compact subsetKof (K(Y),h) such that
Lager Image
Definition 3.2. Let
Lager Image
be a sequence of fuzzy random variables.
  • (i)
Lager Image
  • is said to be level-wise independent if for each α ∈ [0,1], the sequence
Lager Image
  • of random sets is independent.
  • (ii)
Lager Image
  • is said to be independent if the sequence
Lager Image
  • of σ-fields is independent, where
Lager Image
  • is the smallest σ-field which
Lager Image
  • is measurable for all α ∈ [0, 1].
(iii)
Lager Image
  • is said to be tight if for each ε > 0, there exists a compact subsetKof (K(Y),h) such that
Lager Image
  • (iv)
Lager Image
  • is said to be strongly tight if for each ε > 0, there exists a compact subsetKof (F(Y),d∞) such that
Lager Image
  • (v)
Lager Image
  • is said to be compactly uniformly integrable (CUI) if for each ε > 0 there exists a compact subsetKof (K(Y),h) such that
Lager Image
  • (vi)
Lager Image
  • is said to be strong-compactly uniformly integrable (SCUI) if for each ε > 0 there exists a compact subsetKof (F(Y),d∞) such that
Lager Image
It is trivial that strong-compactly uniform integrability (resp. strong tightness) implies compactly uniform integrability (resp. tightness). But, the converse is not true even though Y is finite dimensional.
First, we establish weak law of large numbers for weighted sums of strong-compactly uniformly integrable fuzzy random variables.
Theorem 3.3. Let
Lager Image
be a sequence of integrably bounded fuzzy random variables and let {λ ni } be a double array of real numbers satisfying
Lager Image
Then
Lager Image
if and only if for each α ∈ [0,1],
Lager Image
To prove the above theorem, we need some lemmas obtained by Kim (submitted) which is based on the characterization of relatively compact subsets of ( F ( Y ), d ) established by Greco and Moschen [21] . For easy references, we list them without proof.
Lemma 3.4. Let K be a relatively compact subset of ( F ( Y ), d ). Then
Lager Image
is also relatively compact in ( F ( Y ), d ).
Recall that we can define the concept of convexity on F ( Y ) as in the case of a vector space even though F ( Y ) is not a vector space. That is, K F ( Y ) is said to be convex if λ u ⊕(1−λ) v K whenever u , v K and 0 ≤ λ ≤ 1. Also, the convex hull co ( K ) of K is defined to be the intersection of all convex sets that contains K . Then we can easily show that co ( K ) is equal to the family of consisting of all fuzzy sets in the form λ 1 u 1 ⊕ … ⊕ λ kuk , where u 1 ,..., uk are any elements of K , λ 1 ,...,λ k are nonnegative real numbers satisfying
Lager Image
Lemma 3.5. Let K be a relatively compact subset of ( F ( Y ), d ). Then co ( K ) is also relatively compact in ( F ( Y ), d ).
For a fixed partition π :0 = α 0 < α 1 < … < α r = 1 of [0,1], we define
Lager Image
Then it follows that
Lager Image
From this fact, we can prove easily that
  • gπ(u⊕v)=gπ(u) ⊕gπ(v) andgπ(λu)= λgπ(u).
Lemma 3.6. Let K be a relatively compact subset of ( F ( Y ), d ). Then for each natural number m , there exists a partition π m of [0,1] such that
Lager Image
We are now in a position to prove the main theorem.
Proof of Theorem 3. The necessity is trivial. To prove the sufficiency, We can assume that C = 1 without loss of generality. Let ε > 0 and 0 < δ < 1 be given. By strong-compactly uniform integrability of
Lager Image
, we can choose a compact subset K of ( F ( Y ), d ) such that
Lager Image
Without loss of generality, we may assume that
Lager Image
K is convex and symmetric (i.e., (−1) u K if u K ), and that K contains
Lager Image
for all u K by lemmas 4 and 5.
By lemma 6, we choose a partition π m :0 = α m ,0 < α m ,1 < … < α m,rm of [0, 1] such that
Lager Image
Now we denote
Lager Image
Then by assumptions of K and λ ni , we have
Lager Image
Thus by (2),
Lager Image
Then we have
Lager Image
Hence we obtain
Lager Image
This implies that
Lager Image
For (I), we first note that
Lager Image
And so
Lager Image
Now for (II), since
Lager Image
we have
Lager Image
for sufficiently large n by our assumption. This completes the proof.
Corollary 3.7. Let { Xn } be a sequence of strongly tight fuzzy random variables such that
Lager Image
Then
Lager Image
if and only if for each α ∈ [0,1],
Lager Image
By applying Theorem 3, we can obtain WLLN for level-wise independent case.
Theorem 3.8. Let
Lager Image
be a sequence of level-wise independent and strong-compactly uniformly integrable fuzzy random variables. Then for any Toeplitz sequence {λ ni } satisfying
Lager Image
for some γ > 0,
Lager Image
Proof. Let ε > 0 and 0 < δ < 1 be given and K be a compact subset of ( F ( Y ), d ) such that
Lager Image
Lager Image
Let us denote
Lager Image
Then since
Lager Image
we have that
Lager Image
For (I), we note that for each α ∈ [0,1], the sequence { L α Ũn }of random sets is independent and tight. Since (4) implies
Lager Image
we have that by Corollary 3.2 of Taylor and Inoue [22] ,
Lager Image
By Corollary 7, this implies that (I) → 0 as n → ∞.
Now for (II), since
Lager Image
we have that
Lager Image
Thus for large n ,
Lager Image
which completes the proof.
Corollary 3.9. Let
Lager Image
be a sequence of level-wise independent and strongly tight fuzzy random variables such that
Lager Image
Then for any Toeplitz sequence {λ ni } satisfying
Lager Image
for some γ > 0,
Lager Image
Unfortunately, the following example shows that a sequence of identically distributed fuzzy random variables may not be strong-compactly uniformly integrable.
Example. Let Y = R . For 0 < λ < 1, we define
Lager Image
Then
Lager Image
and so d ( u λ , u δ )= 1 for λ ≠ δ.
Now we let Ω =(0,1), A = the Lebesque σ-field and P be the Lebesgue measure. and let
Lager Image
be a sequence of identically distributed fuzzy random variables with
Lager Image
defined by
Lager Image
Suppose that 0 < ε < 1 and that there is a compact subset K of ( F ( R ), d ) such that
Lager Image
Then K necessarily contains a set of the form
  • KJ= {uλ: λ ∈J},
where P ( J ) > 1 − ε. But this is impossible because KJ contains a sequence { u λ n : λ n J } which does not have any convergent subsequence.
The above example implies that Theorem 3 cannot be applied for identically distributed fuzzy random variables. Guan and Li [14] gave an WLLN for weighted sums of level-wise independent fuzzy random variables under the assumption that
Lager Image
is convergent. The next theorem is slightly different from the result of Guan and Li [14] .
Theorem 3.10. Let
Lager Image
be a sequence of integrably bounded fuzzy random variables such that for some v F ( Y ),
Lager Image
Then
Lager Image
if and only if for each α ∈ [0,1],
Lager Image
and
Lager Image
Proof. To prove the sufficiency, it suffices to prove that
Lager Image
Let
Lager Image
and let ε > 0 be given. By Lemma 4 of Guan and Li [8] , there exists a partition 0 = α 0 < α 1 < … < α r = 1 such that
Lager Image
Then by our assumption, we can find a natural number N such that
Lager Image
First we note that if A 1 A A 2 and B 1 B B 2 , then
  • h(A,B) ≤ max[h(A1,B2),h(A2,B1)].
If 0 < α ≤ 1, then α k −1 < α ≤ α k for some k . Since
Lager Image
we have that for n N ,
Lager Image
Thus for n N ,
Lager Image
Therefore, by assumption we obtain
Lager Image
This completes the proof.
Corollary 3.11. Let
Lager Image
be a sequence of identically distributed fuzzy random variables with
Lager Image
and {λ ni } be a double sequence of real numbers satisfying
Lager Image
Then
Lager Image
if and only if for each α ∈ [0,1]
Lager Image
and
Lager Image
Proof. The necessity is trivial. To prove the sufficiency, we note that
Lager Image
Since
Lager Image
the desired result follows immediately.
4. Conclusions
In this paper, we obtained two types of necessary and sufficient conditions under which weak laws of large numbers for weighted sums of fuzzy random variables hold. One is the case of strong-compactly uniformly integrable fuzzy random variables. The other is the case that the weighted averages of its expectations converge. The former includes a strongly tight case and the latter contains the identically distributed case. We also provided WLLN for weighted sums of level-wise independent and strong-compactly uniformly integrable (or strongly tight) fuzzy random variables.
It remains an open problem whether we can obtain a generalization for the above WLLN to the case of compactly uniform integrability.
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