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
A norm of
It is well-known that K(
The addition and scalar multiplication on K(
A ？ B = {a + b : a ∈ A, b ∈ B}, λA = {λa : a ∈ A}
for
The convex hull and closed convex hull of
, respectively. If
∈ K(
Let F(
(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
Then it follows immediately that
The linear structure on F(
for
denotes the indicator function of {0}.
Then it is known that for each α ∈ [0,1],
Recall that a fuzzy subset
u(λx+(1？λ)y) ≥ min(u(x),u(y)) for x,y ∈ Y and λ ∈ [0,1].
The convex hull of
co(u) = inf{v : v is convex and v ≥ u}.
Then it is known that for each α ∈ [0,1],
If
of
Then it is well-known that
for each α ∈ [0,1] and
The uniform metric
It is well-known that (F(
Throughout this paper, let (Ω,
E(X)= {E(ξ) : ξ ∈ L(Ω,Y) and ξ(ω) ∈ X(ω)a.s.},
where
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
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
where
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 {
(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
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
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(
Lemma 3.4. Let
is also relatively compact in (F(
Recall that we can define the concept of convexity on F(
Lemma 3.5. Let
For a fixed partition π :0 = α_{0} < α_{1} < … < α
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
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
, we can choose a compact subset
Without loss of generality, we may assume that
for all
By lemma 6, we choose a partition π
Now we denote
Then by assumptions of
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
Corollary 3.7. Let {
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
for some γ > 0,
Let us denote
Then since
we have that
For (I), we note that for each α ∈ [0,1], the sequence {
we have that by Corollary 3.2 of Taylor and Inoue [22],
By Corollary 7, this implies that (I) → 0 as
Now for (II), since
we have that
Thus for large
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
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
Then
and so
Now we let Ω =(0,1),
be a sequence of identically distributed fuzzy random variables with
defined by
Suppose that 0 < ε < 1 and that there is a compact subset
Then
KJ = {uλ : λ ∈ J},
where
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
Then
if and only if for each α ∈ [0,1],
and
Let
and let ε > 0 be given. By Lemma 4 of Guan and Li [8], there exists a partition 0 = α_{0} < α_{1} < … < α
Then by our assumption, we can find a natural number
First we note that if
h(A, B) ≤ max[h(A1,B2),h(A2,B1)].
If 0 < α ≤ 1, then α
we have that for
Thus for
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
Then
if and only if for each α ∈ [0,1]
and
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.