In this paper, we will obtain Marcinkiewicz's type limit laws for fuzzy random sets as follows : Let {
X_{n}

n
≥ 1} be a sequence of independent identically distributed fuzzy random sets and
with 1≤
r
≤ 2. Then the following are equivalent:
a.s. in the metric
ρ_{p}
if and only if
in probability in the metric
ρ_{p}
if and only if
in
L
_{1}
if and only if
in
L
_{r}
where
.
AMS Mathematics Subject Classification : 60F05. 60G05.
1. Introduction
The study of the fuzzy random sets, defined as measurable mappings on a probability space, was initiated by Kwakernaak
[12]
where useful basic properties were developed. Puri and Ralescu
[9]
used the concept of fuzzy random variables in generating results for random sets to fuzzy random sets. Kruse
[8]
proved a strong law of large numbers for independent identically distributed fuzzy random variables. Artstein and Vitale
[1]
proved a strong law of large numbers(SLLN) for
R^{p}
valued random sets and Cressie
[3]
proved a SLLN for some paticular class of
R^{p}
valued random sets. Using Rådstrom embedding(e.g. Rådstrom
[14]
), Puri and Ralescu
[12]
proved a SLLN for Banach space valued random sets and they also proved SLLN for fuzzy random sets, which generalized all of previous SLLN for random sets. In recent year, Joo, Kim and Kwon
[6]
proved Chung’s type law of large numbers for fuzzy random variables and Kwon and Shim
[11]
obtained a uniform strong law of large numbers for partial sum processes of fuzzy random sets. In this paper we obtain Marcinkiewicz’s type laws for fuzzy random sets in the Euclidean space under the assumption that WLLN holds. The proofs of the results are based heavily on isometrical embeddings of the fuzzy sample spaces, endowed with
L_{p}
metrics, into
L_{p}
spaces. Our results give the fuzzy version of Marcinkiewicz’s type law of large numbers in general Banach spaces.
2. Preliminaries
Let
K
(
R^{n}
) (
K
_{c}
(
R^{n}
) ) be the collection of nonempty compact (and convex ) subsets of Euclidean space
R^{n}
. The set can be viewed as a linear structure induced by the scalar multiplication and the Minkowski addition, that is
for all
A
,
B
∈
K
(
R^{n}
) and λ ∈
R
. If
d
is the Hausdoff metric on
K
(
R^{n}
) which, for
A
,
B
∈
K
(
R^{n}
) , is given by
where · denotes the Euclidean norm. Then (
K
(
R^{n}
) ,
d
) is a complete separable metric space
[4
,
10]
.
A fuzzy set of
R^{n}
is a mapping
A
:
R^{n}
→ [0, 1]. We will denote by A
_{α}
the
α
level set of
A
(that is
A_{α}
= {
x
∈
R^{n}
:
A
(
x
) ≥
α
} ) for all
α
∈ (0, 1] and by
A
_{0}
the closure of the support of
A
(that is
A
_{0}
=
cl
{
x
∈
R^{n}
:
A
(
x
) > 0} ).
Let
F_{c}
(
R^{n}
)(
F_{coc}
(
R^{n}
)) be the class of the fuzzy sets A satisfying the following conditions

(1)A1≠ ∅,

(2)A0is compact, and

(3)Ais upper semi continuous

((4)Aαis convex for allα∈ [0, 1]).
And
is the subset of
F_{c}
(
R^{n}
)(
F_{coc}
(
R^{n}
)) with bounded support.
Given a measurable space (Ω,
A
) and the metric space (
K
(
R^{n}
) ,
d
), a random set (or as a random compact set) is associated with a Borel measurable mapping
X
: Ω →
K
(
R^{n}
) . If
X
: Ω →
K
(
R^{n}
) is a setvalued mapping, then
X
is a random set if and only if
X
^{−1}
(
C
) = {
ω
∈ Ω :
X
(
ω
) ∩
C
≠ ∅} ∈
A
for all
C
∈
K
(
R^{n}
) .
If
X
is a random set, the mapping denoted by ∥
X
∥
_{d}
and defined by
for all
ω
∈ Ω, is a random variable, where
is the fuzzy set where
and
otherwise.
A support function of a nonvoid bounded subset
K
of
R^{n}
is defined by
where <
x
,
y
> denotes the standard scalar product of the vectors
x
and
y
. Support functions
s_{K}
are uniquely associated with the subsets
K
∈
K
_{c}
(
R^{n}
) and preserve addition and nonnegative scalar multiplication when we restricted ourselves to
K
(
R^{n}
), i.e.
Now we endow
F_{c}
(
R^{n}
) with the initial topology generated by the mappings
then the topology mentioned above enables us to introduce a measurability concept for defining fuzzy random variable. We call a mapping
X
: Ω →
F_{c}
(
R^{n}
) fuzzy random variable over (Ω,
A
,
μ
) if it is
A
measurable over the initial topology.
For a real number
p
≥ 1 and
A
,
B
∈
F_{c}
(
R^{n}
) , define
and
where
denotes the unit Lebesgue measure on the unit sphere in
R^{n}
. Then
d_{p}
(
ρ_{p}
) becomes a separable metric on
with the relation
ρ_{p}
≤
d_{p}
which induce the same topology
Now consider
L_{p}
([0, 1]×
S
^{n−1}
), the
L_{p}
space with respect to [0, 1]×
S
^{n−1}
, the obvious product
σ
algebra and the product measure
. Then under the
L_{p}
norm ∥∥
_{p}
we obtain
L_{p}
([0, 1]×
S
^{n−1}
) as a separable Banach space. Next we can embed
isometrically isomorphic into
L_{p}
([0, 1]×
S
^{n−1}
) as a positive cone (for details see
[5
,
7]
). Embedding
into
L_{p}
([0, 1]×
S
^{n−1}
), we draw a convergence theorem in Banach space. For 1 ≤
p
< ∞,
L_{p}
([0, 1] ×
S
^{n−1}
) is so called separable Banach space of type min(
p
, 2). It is known that separable Banach spaces of type 2 is exactly those separable Banach space where the classical strong law of large numbers for independent nonidentically distributed random variables holds.
3. Main Results
To prove the main theorem we will need the following lemmas. Lemma 1 connects two metric spaces
and
L_{p}
([0, 1] ×
S
^{n−1}
) isometrically.
Lemma 3.1.
Let
1 ≤
p
< ∞
be fixed
.
Then j
:
→
L_{p}
([0, 1] ×
S
^{n−1}
)
by A
defines an injection mapping satisfying

(1)∥j(A) −j(B)∥p=ρp(A,B)

(2)j(A+B) =j(A) +j(B)

(3)j(λA) = λj(A)

for any
The following is a generalization of a classical result
[15, p. 127128]
.
Lemma 3.2.
Let
{
X_{n}

n
≥ 1}
be a sequence of fuzzy random sets stochastically dominated by X with
for
0 <
r
< ∞,
that is
,
for any t
> 0,
.
Then
Proof. .
Notice that
is a sequence of random variables stochastically dominated by
. Now apply Stout’s result.
Lemma 3.3
(
[5]
).
Let
{
X_{k}
1 ≤
k
≤
n
}
be

valued independent random variables
.
Let
Then
Lemma 3.4
(
[5]
).
Let
{
X_{k}
1 ≤
k
≤
n
}
be independent

valued random variables with
for k
= 1, 2, · · ·
n and
1 ≤
r
≤ 2.
Then we have
where C_{r} is a positive constant depending only on r ; if r
= 2
then it is possible to take C
_{2}
= 4.
Theorem 3.5.
Let
{
X_{n}

n
≥ 1}
be a sequence of independent identically distributed

valued fuzzy random variables with
for
1≤
r
≤2
and let
.
Then the following are equivalent:

(i)a.s. in the metric ρp;

(ii)in probability in the metric ρp;

(iii)in L1

(iv)in Lr
Proof
. Let
j
:
→
L_{p}
([0, 1]×
S
^{n−1}
) be an isometry. Then {
j
◦
X_{n}

n
≥ 1} be a sequence of independent identically distributed random element in a Banach space
L_{p}
([0, 1] ×
S
^{n−1}
). Since
in what follow we use
X
and
j
◦
X
interchangeably.
First we show that (
i
) ⇔ (
ii
) ⇔ (
iii
) in
L^{1}
. Since
, we have
Hence it is enough to show that
Since
by lemma 1, these equivalence hold by applying Theorem 5 in
[5]
to
Now it remains to show that (iii) ⇒ (iv). Assume that
in
L
_{1}
.
Then now
Thus it is enough to show that
From Lemma 4
By a standard calculation, we have
Thus the proof is completed.
Remark 3.1.
(1) For i.i.d real valued random variables, Pyke and Root
[13]
showed that
(2) For i.i.d. Bvalued random variables with
E
∥
X
_{1}
∥ < ∞ for 1 ≤
r
< 2, Choi and Sung
[2]
showed that
BIO
JoongSung Kwon received his Ph.D at University of Washington. Since 1992 he has been a professor at Sunmoon University. His research interest is stochastic limit theory and fuzzy set theory.
Department of Mathematics, Sun Moon University, 100 Kalsanri, Tangieongmyeon, Asansi, Choongnam, 336840, Korea.
email: jskwon@sunmoon.ac.kr
HongTae Shim received Ph. D from the University of WisconsinMilwaukee. His research interests center on wavelet theories, Sampling theories and Gibbs' phenomenon for series of special functions.
Department of Mathematics, Sun Moon University, 100 Kalsanri, Tangieongmyeon, Asansi, Choongnam, 336840, Korea.
email: hongtae@sunmoon.ac.kr
Choi B.D.
,
Sung S.H.
(1988)
On Chung's strong law of large numbers in general Banach spaces
Bull. Austral. Math. Soc.
37
93 
100
DOI : 10.1017/S0004972700004184
Cressie N.
(1978)
A strong limit theorem for random sets
Suppl. Adv. Appl. Prob.
10
36 
46
DOI : 10.2307/1427005
Debreu G.
1967
Integration of correspondences
University of California Press
Berkely
in: Proc. 5th Berkely Symp. on Mathematical Statistics and Probability
Vol. II, Part I
351 
372
Hong D.H.
,
Moon Eunho
,
Kim J.D.
(2010)
Convexity and semicontinuity of fuzzy mappings using the support function
J. Appl. Math. & Informatics
28
1419 
1430
Joo S.Y.
,
Kim Y.K.
,
Kwon J.S.
(2005)
On Chung's type law of large numbers for fuzzy random variables
Stat, and Probab Letters
74
67 
75
DOI : 10.1016/j.spl.2005.04.030
Kwon J.S.
,
Shim H.T.
(2011)
Overviews on limit concrepts of a sequence of fuzzy numbers I
J. Appl. Math. & Informatics
29
1017 
1025
Kwon J.S.
,
Shim H.T.
(2012)
A uniform strong law of large numbers for partial sum processes of fuzzy random sets
J. Appl. Math. & Informatics
30
647 
653
Puri M.L.
,
Ralescu D.A.
(1983)
Strong law of large numbers for Banach space valued random sets
Ann. Probab.
11
222 
224
DOI : 10.1214/aop/1176993671
Rådstrom H.
(1952)
An embedding theorem for spaces of convex sets
Proc. Amer. Math. Soc.
3
165 
169
DOI : 10.2307/2032477
Stout W.F.
1974
Almost sure convergence
Academic press
New York