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

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 30, 2013

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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.
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:
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;
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
_{α}
u
∈
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
_{α}
u
⊕
L
_{α}
v
and
L
_{α}
(λ
u
)= λ
L
_{α}
u
.
Recall that a fuzzy subset
u
of
Y
is said to be convex if
The convex hull of
u
is defined by
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
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]
).
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
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.
Definition 3.2.
Let
be a sequence of fuzzy random variables.
(iii)
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 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
_{k}u_{k}
, where
u
_{1}
,...,
u_{k}
are any elements of
K
, λ
_{1}
,...,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 = α
_{0}
< α
_{1}
< … < α
_{r}
= 1 of [0,1], we define
Then it follows that
From this fact, we can prove easily that
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}
Ũ_{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
where
P
(
J
) > 1 − ε. But this is impossible because
K_{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 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 = α
_{0}
< α
_{1}
< … < α
_{r}
= 1 such that
Then by our assumption, we can find a natural number
N
such that
First we note that if
A
_{1}
⊂
A
⊂
A
_{2}
and
B
_{1}
⊂
B
⊂
B
_{2}
, then
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.

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 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
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- A⊕B= {a+b:a∈A,b∈B}, λA= {λa:a∈A}

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- (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.

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- u(λx+(1–λ)y) ≥ min(u(x),u(y)) forx,y∈Yand λ ∈ [0,1].

- co(u) = inf{v:vis convex andv≥u}.

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3. Main Results

Throughout this paper, let (Ω,
- E(X)= {E(ξ) : ξ ∈L(Ω,Y) and ξ(ω) ∈X(ω)a.s.},

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- (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

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- (i)

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

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- of random sets is independent.
- (ii)

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- is said to be independent if the sequence

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- of σ-fields is independent, where

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- is the smallest σ-field which

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- is measurable for all α ∈ [0, 1].

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- is said to be tight if for each ε > 0, there exists a compact subsetKof (K(Y),h) such that

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- (iv)

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- is said to be strongly tight if for each ε > 0, there exists a compact subsetKof (F(Y),d∞) such that

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- (v)

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- is said to be compactly uniformly integrable (CUI) if for each ε > 0 there exists a compact subsetKof (K(Y),h) such that

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- (vi)

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- is said to be strong-compactly uniformly integrable (SCUI) if for each ε > 0 there exists a compact subsetKof (F(Y),d∞) such that

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- gπ(u⊕v)=gπ(u) ⊕gπ(v) andgπ(λu)= λgπ(u).

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- KJ= {uλ: λ ∈J},

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- h(A,B) ≤ max[h(A1,B2),h(A2,B1)].

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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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Citing 'Weak laws of large numbers for weighted sums of Banach space valued fuzzy random variables
'

@article{ E1FLA5_2013_v13n3_215}
,title={Weak laws of large numbers for weighted sums of Banach space valued fuzzy random variables}
,volume={3}
, url={http://dx.doi.org/10.5391/IJFIS.2013.13.3.215}, DOI={10.5391/IJFIS.2013.13.3.215}
, number= {3}
, journal={International Journal of Fuzzy Logic and Intelligent Systems}
, publisher={Korean Institute of Intelligent Systems}
, author={Kim, Yun Kyong}
, year={2013}
, month={Sep}