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.

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

A norm of A ∈ K(Y) is defined by

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

, 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

∈ 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) u is normal, i.e., there exists x ∈ Y such that u(x)= 1;

(ii) u is upper-semicontinuous;

(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 of Y, the α-level set of u is defined by

Then it follows immediately that u ∈ F(Y) if and only if L_αu ∈ K(Y) for each a ∈ [0,1]: If we denote cl{x ∈ Y : u(x) > α} by L_α+u, then

The linear structure on F(Y) is also defined as usual;

for u,v ∈ F(Y) and λ ∈ R, where

denotes the indicator function of {0}.

Then it is known that for each α ∈ [0,1], L_α (u ？ v)= L_αu ？ L_αv and L_α(λu)= λL_αu.

Recall that a fuzzy subset u of Y is said to be convex if

u(λx+(1？λ)y) ≥ min(u(x),u(y)) for x,y ∈ Y and λ ∈ [0,1].

The convex hull of u is defined by

co(u) = inf{v : v is convex and v ≥ u}.

Then it is known that for each α ∈ [0,1], L_αco(u)= co(L_αu).

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

of u is defined by

Then it is well-known that

for each α ∈ [0,1] and

The uniform metric d_∞ and norm ||？|| on F(Y) as usual;

It is well-known that (F(Y),d_∞) is complete but is not separable (see Klement et al. [8]).

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

is called a fuzzy random variable (or fuzzy random set) if for each

is a random set. It is well-known 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 by

For 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 satisfying

where C > 0 is a constant not depending on n.

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. Let

be a sequence of fuzzy random variables.

(i)

is said to be level-wise independent if for each α ∈ [0,1], the sequence

of random sets is independent.

(ii)

is said to be independent if the sequence

of σ-fields is independent, where

is the smallest σ-field 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 strong-compactly uniformly integrable (SCUI) if for each ε > 0 there exists a compact subset K of (F(Y),d∞) such that

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

be a sequence of integrably bounded fuzzy random variables and let {λ_ni} be a double array of real numbers satisfying

Then

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. Let K be a relatively compact subset of (F(Y),d_∞). Then

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 λu？(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 λ₁u₁ ？ … ？ λ_ku_k, where u₁,...,u_k are any elements of K, λ₁,...,λ_k are nonnegative real numbers satisfying

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 = α₀ < α₁ < … < α_r = 1 of [0,1], we define

Then 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. 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

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

, we can choose a compact subset K of (F(Y),d_∞) such that

Without loss of generality, we may assume that

K is convex and symmetric (i.e., (？1)u ∈ K if u ∈ K), and that K contains

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

Now we denote

Then by assumptions of K and λ_ni, we have

Thus 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 that

Then

if and only if for each α ∈ [0,1],

By applying Theorem 3, we can obtain WLLN for level-wise independent case.

Theorem 3.8. Let

be a sequence of level-wise independent and strong-compactly uniformly integrable fuzzy random variables. Then for any Toeplitz sequence {λ_ni} satisfying

for some γ > 0,

Proof. Let ε > 0 and 0 < δ < 1 be given and K be a compact subset of (F(Y),d_∞) such that

Let 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) implies

we 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. Let

be a sequence of level-wise independent and strongly tight fuzzy random variables such that

Then for any Toeplitz sequence {λ_ni} satisfying

for some γ > 0,

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

Then

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

be 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 that

Then K necessarily contains a set of the form

KJ = {uλ : λ ∈ J},

where P(J) > 1 ？ ε. But this is impossible because K_J contains a sequence {u_λn : 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

is convergent. The next theorem is slightly different from the result of Guan and Li [14].

Theorem 3.10. Let

be 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 that

Let

and let ε > 0 be given. By Lemma 4 of Guan and Li [8], there exists a partition 0 = α₀ < α₁ < … < α_r = 1 such that

Then by our assumption, we can find a natural number N such that

First we note that if A₁ ⊂ A ⊂ A₂ and B₁ ⊂ B ⊂ B₂, 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

we have that for n ≥ N,

Thus for n ≥ N,

Therefore, by assumption we obtain

This completes the proof.

Corollary 3.11. Let

be a sequence of identically distributed fuzzy random variables with

and {λ_ni} be a double sequence of real numbers satisfying

Then

if and only if for each α ∈ [0,1]

and

Proof. The necessity is trivial. To prove the sufficiency, we note that

Since

the desired result follows immediately.

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.