Weak laws of large numbers for weighted sums of Banach space valued fuzzy random variables
 Author: Kim Yun Kyong
 Organization: Kim Yun Kyong
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 13, Issue3, p215~223, 25 Sep 2013

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 uppersemicontinuous fuzzy sets with compact support in a separable real Banach space. First, we give weak laws of large numbers for weighted sums of strongcompactly 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.

KEYWORD
Fuzzy sets , Random sets , Fuzzy random variables , Weak law of large numbers , Compactly uniform integrability , Tightness , Weighted sum.

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 convexcompactly uniformly integrable fuzzy random variables taking values in the space of fuzzy numbers in a finitedimensional 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 levelcontinuous 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 uppersemicontinuous fuzzy sets with compact support in a real separable Banach space. First, we give WLLN for the weighted sum of strongcompactly 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 letK (Y ) denote the family of all nonempty compact subsets ofY . Then the spaceK (Y ) is metrizable by the Hausdorff metrich defined byA norm of
A ∈K (Y ) is defined byIt is wellknown that
K (Y ) is complete and separable with respect to the Hausdorff metrich (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 byco (A ) and, respectively. If
dim (Y ) < ∞ andA ∈K (Y ), thenco (A ) ∈K (Y ). But ifdim (Y )= ∞, it is wellknown thatco (A ) may not be an element ofK (Y ) even thoughA ∈K (Y ), but∈
K (Y ) ifA ∈K (Y ).Let
F (Y ) denote the family of all fuzzy setsu :Y → [0,1] with the following properties;(i) u is normal, i.e., there exists x ∈ Y such that u(x)= 1;
(ii) u is uppersemicontinuous;
(iii) supp u = cl{x ∈ Y : u(x) > 0} is compact, where cl(A) denotes the closure of A in Y.
For a fuzzy subset
u ofY , the αlevel set ofu is defined byThen it follows immediately that
u ∈F (Y ) if and only ifL _{α}u ∈K (Y ) for each a ∈ [0,1]: If we denotecl {x ∈Y :u (x ) > α} byL _{α}+u , thenThe linear structure on
F (Y ) is also defined as usual;for
u ,v ∈F (Y ) and λ ∈R , wheredenotes 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 andL _{α}(λu )= λL _{α}u .Recall that a fuzzy subset
u ofY is said to be convex ifu(λx+(1？λ)y) ≥ min(u(x),u(y)) for x,y ∈ Y and λ ∈ [0,1].
The convex hull of
u is defined byco(u) = inf{v : v is convex and v ≥ 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 andu ∈F (Y ), thenco (u ) ∈F (Y ). But ifY is infinite dimensional space, it may not be true. So we need the notion of the closed convex hull ofu . The closed convex hullof
u is defined byThen it is wellknown that
for each α ∈ [0,1] and
The uniform metric
d _{∞} and norm ？ onF (Y ) as usual;It is wellknown 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 setvalued functionX : Ω → (K (Y ),h ) is called a random set if it is measurable. A random setX is said to be integrably bounded ifE X  < ∞. The expectation of integrably bounded random setX is defined byE(X)= {E(ξ) : ξ ∈ L(Ω,Y) and ξ(ω) ∈ X(ω)a.s.},
where
L (Ω,Y ) denotes the class of allY valued random variables ξ such thatE ξ < ∞.A fuzzy set valued function
is called a fuzzy random variable (or fuzzy random set) if for each
is a random set. It is wellknown that if
is measurable, then
is a fuzzy random variable. But the converse is not true (For details, see Colubi et al. [18], Kim [19]).
A fuzzy random set
is said to be integrably bounded if
The expectation of integrably bounded fuzzy random variable
is a fuzzy subset
of
Y defined byFor more details for expectations of random sets and fuzzy random variables, the readers may refer to Li et al. [20].
Let
be a sequence of integrably bounded fuzzy random variables and {λ
_{ni} } be a double array of real numbers that not necessarily Toeplitz but satisfyingwhere
C > 0 is a constant not depending onn .The problem that we will consider is to establish sufficient conditions for
where
denotes the closed convex hull of
To this end, we need the concepts of tightness and compact uniform integrability for a sequence of fuzzy random variables.
Definition 3.1. Let {X_{n} } be a sequence of random sets.(i) {Xn} is said to be tight if for each ε > 0, there exists a compact subset K of (K(Y), h) such that
P(Xn ？ K) < ε for all n.
(ii) {Xn} is said to be compactly uniformly integrable(CUI) if for each ε > 0, there exists a compact subset K of (K(Y),h) such that
Definition 3.2. Letbe a sequence of fuzzy random variables.
(i)
is said to be levelwise independent if for each α ∈ [0,1], the sequence
of random sets is independent.
(ii)
is said to be independent if the sequence
of σfields is independent, where
is the smallest σfield which
is measurable for all α ∈ [0, 1].
(iii)
is said to be tight if for each ε > 0, there exists a compact subset K of (K(Y), h) such that
(iv)
is said to be strongly tight if for each ε > 0, there exists a compact subset K of (F(Y),d∞) such that
(v)
is said to be compactly uniformly integrable (CUI) if for each ε > 0 there exists a compact subset K of (K(Y),h) such that
(vi)
is said to be strongcompactly uniformly integrable (SCUI) if for each ε > 0 there exists a compact subset K of (F(Y),d∞) such that
It is trivial that strongcompactly 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 strongcompactly uniformly integrable fuzzy random variables.
Theorem 3.3. Letbe a sequence of integrably bounded fuzzy random variables and let {λ
_{ni} } be a double array of real numbers satisfyingThen
if and only if for each α ∈ [0,1],
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. LetK be a relatively compact subset of (F (Y ),d _{∞}). Thenis 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 thoughF (Y ) is not a vector space. That is,K ⊂F (Y ) is said to be convex if λu ？(1？λ)v ∈K wheneveru ,v ∈K and 0 ≤ λ ≤ 1. Also, the convex hullco (K ) ofK is defined to be the intersection of all convex sets that containsK . Then we can easily show thatco (K ) is equal to the family of consisting of all fuzzy sets in the form λ_{1}u _{1} ？ … ？ λ_{k}u_{k} , whereu _{1},...,u_{k} are any elements ofK , λ_{1},...,λ_{k} are nonnegative real numbers satisfyingLemma 3.5. LetK be a relatively compact subset of (F (Y ),d _{∞}). Thenco (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 defineThen it follows that
From this fact, we can prove easily that
gπ(u ？ v)= gπ(u) ？ gπ(v) and gπ(λu)= λgπ(u).
Lemma 3.6. LetK be a relatively compact subset of (F (Y ),d _{∞}). Then for each natural numberm , there exists a partition π_{m} of [0,1] such thatWe 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 thatC = 1 without loss of generality. Let ε > 0 and 0 < δ < 1 be given. By strongcompactly uniform integrability of, we can choose a compact subset
K of (F (Y ),d _{∞}) such thatWithout loss of generality, we may assume that
K is convex and symmetric (i.e., (？1)u ∈K ifu ∈K ), and thatK containsfor 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 thatNow we denote
Then by assumptions of
K and λ_{ni} , we haveThus by (2),
Then we have
Hence we obtain
This implies that
For (I), we first note that
And so
Now for (II), since
we have
for sufficiently large
n by our assumption. This completes the proof.Corollary 3.7. Let {X_{n} } be a sequence of strongly tight fuzzy random variables such thatThen
if and only if for each α ∈ [0,1],
By applying Theorem 3, we can obtain WLLN for levelwise independent case.
Theorem 3.8. Letbe a sequence of levelwise independent and strongcompactly uniformly integrable fuzzy random variables. Then for any Toeplitz sequence {λ
_{ni} } satisfyingfor some γ > 0,
Proof. Let ε > 0 and 0 < δ < 1 be given andK be a compact subset of (F (Y ),d _{∞}) such thatLet us denote
Then since
we have that
For (I), we note that for each α ∈ [0,1], the sequence {
L _{α}？_{n} }of random sets is independent and tight. Since (4) implieswe have that by Corollary 3.2 of Taylor and Inoue [22],
By Corollary 7, this implies that (I) → 0 as
n → ∞.Now for (II), since
we have that
Thus for large
n ,which completes the proof.
Corollary 3.9. Letbe a sequence of levelwise independent and strongly tight fuzzy random variables such that
Then for any Toeplitz sequence {λ
_{ni} } satisfyingfor some γ > 0,
Unfortunately, the following example shows that a sequence of identically distributed fuzzy random variables may not be strongcompactly uniformly integrable.
Example. LetY =R . For 0 < λ < 1, we defineThen
and so
d _{∞}(u _{λ},u _{δ})= 1 for λ ≠ δ.Now we let Ω =(0,1),
A = the Lebesque σfield andP be the Lebesgue measure. and letbe a sequence of identically distributed fuzzy random variables with
defined by
Suppose that 0 < ε < 1 and that there is a compact subset
K of (F (R ),d _{∞}) such thatThen
K necessarily contains a set of the formKJ = {uλ : λ ∈ J},
where
P (J ) > 1 ？ ε. But this is impossible becauseK_{J} 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 levelwise independent fuzzy random variables under the assumption that
is convergent. The next theorem is slightly different from the result of Guan and Li [14].
Theorem 3.10. Letbe a sequence of integrably bounded fuzzy random variables such that for some
v ∈F (Y ),Then
if and only if for each α ∈ [0,1],
and
Proof. To prove the sufficiency, it suffices to prove thatLet
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 thatThen by our assumption, we can find a natural number
N such thatFirst we note that if
A _{1} ⊂A ⊂A _{2} andB _{1} ⊂B ⊂B _{2}, thenh(A, B) ≤ max[h(A1,B2),h(A2,B1)].
If 0 < α ≤ 1, then α
_{k} _{？1} < α ≤ α_{k} for somek . Sincewe have that for
n ≥N ,Thus for
n ≥N ,Therefore, by assumption we obtain
This completes the proof.
Corollary 3.11. Letbe a sequence of identically distributed fuzzy random variables with
and {λ
_{ni} } be a double sequence of real numbers satisfyingThen
if and only if for each α ∈ [0,1]
and
Proof. The necessity is trivial. To prove the sufficiency, we note thatSince
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 strongcompactly 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 levelwise independent and strongcompactly 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.