Aug 15, 2016 - Taylor and Inoue had proved the strong law of large numbers for independent ... space, which is both the extension of the result in [1] for ...
Advances in Pure Mathematics, 2016, 6, 583-592 Published Online August 2016 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2016.69047
Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in Gα Space Lamei Shen, Li Guan College of Applied Sciences, Beijing University of Technology, Beijing, China Received 7 July 2016; accepted 12 August 2016; published 15 August 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of dH∞ . The results are based on the result of single-valued random variables obtained by Taylor [1] and set-valued random variables obtained by Li Guan [2].
Keywords Laws of Large Numbers, Fuzzy Set-Valued Random Variable, Hausdorff Metric
1. Introduction With the development of set-valued stochastic theory, it has become a new branch of probability theory. And limits theory is one of the most important theories in probability and statistics. Many scholars have done a lot of research in this aspect. For example, Artstein and Vitale in [3] had proved the strong law of large numbers for independent and identically distributed random variables by embedding theory. Hiai in [4] had extended it to separable Banach space. Taylor and Inoue had proved the strong law of large numbers for independent random variable in the Banach space in [5]. Many other scholars also had done lots of works in the laws of large numbers for set-valued random variables. In [2], Li proved the strong laws of large numbers for set-valued random variables in Gα space in the sense of dH metric. As we know, the fuzzy set is an extension of the set. And the concept of fuzzy set-valued random variables is a natural generalization of that of set-valued random variables, so it is necessary to discuss convergence theorems of fuzzy set-valued random sequence. The limits of theories for fuzzy set-valued random sequences are also been discussed by many researchers. Colubi et al. [6], Feng [7] and Molchanov [8] proved the strong laws of large numbers for fuzzy set-valued random variables; Puri and Ralescu [9], Li and Ogura [10] proved converHow to cite this paper: Shen, L.M. and Guan, L. (2016) Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in Gα Space. Advances in Pure Mathematics, 6, 583-592. http://dx.doi.org/10.4236/apm.2016.69047
L. M. Shen, L. Guan
gence theorems for fuzzy set-valued martingales. Li and Ogura [11] proved the SLLN of [12] in the sense of d H∞ by using the “sandwich” method. Guan and Li [13] proved the SLLN for weighted sums of fuzzy setvalued random variables in the sense of d H∞ which used the same method. In this paper, what we concerned are the convergence theorems of fuzzy set-valued sequence in Gα space in the sense of d H∞ . The purpose of this paper is to prove the strong laws of large numbers for fuzzy set-valued random variables in Gα space, which is both the extension of the result in [1] for single-valued random sequence and also the extension in [2] for set-valued random sequence. This paper is organized as follows. In Section 2, we shall briefly introduce some concepts and basic results of set-valued and fuzzy set-valued random variables. In Section 3, I shall prove the strong laws of large numbers for fuzzy set-valued random variables in Gα space, which is in the sense of Hausdorff metric d H∞ .
2. Preliminaries on Set-Valued Random Variables Throughout this paper, we assume that ( Ω, , µ ) is a complete probability space, ( X, ⋅ ) is a real separable Banach space, K ( X ) is the family of all nonempty closed subsets of X , and K b ( X ) ( K k ( X ) ) is the family
of all non-empty bounded closed(compact) subsets of X , and K kc ( X ) is the family of all non-empty compact convex subsets of X . Let A and B be two nonempty subsets of X and let λ ∈ , the set of all real numbers. We define addition and scalar multiplication by
A + B = {a + b : a ∈ A, b ∈ B}
{λ a : a ∈ A}
= λA The Hausdorff metric on K ( X ) is defined by
{
d H ( A, B )= max sup inf a − b , sup inf a − b a∈A b∈B
b∈B a∈A
}
for A, B ∈ K ( X ) . For an A in K ( X ) , let A K = d H ({0} , A ) . The metric space ( K b ( X ) , d H ) is complete, and K bc ( X ) is a closed subset of
(Kb (X ) , dH )
(cf. [14],
Theorems 1.1.2 and 1.1.3). For more general hyperspaces, more topological properties of hyperspaces, readers may refer to the books [15] and [14]. For each A ∈ K ( X ) , define the support function by
(
)
= s x* , A sup x* , a , x* ∈ X* , *
where X is the dual space of X . Let S* denote the unit sphere of X* , C S* as v C = sup x∗ ∈S∗ .
( )
a∈A
the all continuous functions of S* , and the norm is defined
The following is the equivalent definition of Hausdorff metric. For each A, B ∈ K bc ( X ) ,
{(
) (
)
}
d H ( A, B ) = sup s x* , A − s x* , A : x* ∈ S * .
A set-valued mapping F : Ω → K ( X ) is called a set-valued random variable (or a random set, or a multifunction) if, for each open subset O of X , F −1 ( O= ) {ω ∈ Ω : F (ω ) ∩ O ≠ ∅} ∈ . For each set-valued random variable F, the expectation of F, denoted by E [ F ] , is defined by
= E [F ] where
∫Ω fdµ
{
{∫
Ω
}
fdµ : f ∈ S F ,
is the usual Bochner integral in L1 [ Ω, X ] , the family of integrable X -valued random variables,
}
and S F = f ∈ L1 [ Ω; X ] : f (ω ) ∈ F (ω ) , a.e. ( µ ) .
584
L. M. Shen, L. Guan
Let Fk ( X ) denote the family of all functions v : X → [ 0,1] which satisfy the following conditions: 1) The level set v1= { x ∈ X : v ( x )= 1} ≠ ∅ . 2) Each v is upper semicontinuous, i.e. for each α ∈ ( 0,1] , the α level set vα = { x ∈ X : v ( x ) ≥ α } is a closed subset of X . cl { x ∈ X : v ( x ) > 0} is compact. 3) The support set v0+ = A function v in Fk ( X ) is called convex if it satisfies
v ( λ x + (1 − λ ) y ) ≥ min {v ( x ) , v ( y )} , for any x, y ∈ X, λ ∈ ( 0,1] .
Let Fkc ( X ) be the subset of all convex fuzzy sets in Fk ( X ) . It is known that v is convex in the above sense if and only if, for any α ∈ ( 0,1] , the level set vα is a convex subset of X (cf. Theorem 3.2.1 of [16]). For any v ∈ Fk ( X ) , the closed convex hull cov ∈ Fkc ( X ) of v is defined by the relation
( cov )
α
= covα for all α ∈ ( 0,1] .
For any two fuzzy sets ν ,ν , define 1
2
(ν
1
+ν 2
) ( x=)
{
}
sup α ∈ ( 0,1] : x ∈ν α1 + ν α2 ,
for any x ∈ X. Similarly for a fuzzy set ν and a real number λ , define
( λν )( x ) =sup {α ∈ ( 0,1] : x ∈ λν α } , for any x ∈ X. The following two metrics in Fk ( X ) which are extensions of the Hausdorff metric dH are often used (cf. [17] and [18], or [14]): for v1 , v 2 ∈ Fk ( X ) ,
(
)
(
)
(
)
d H∞ v1 , v 2 = sup d H vα1 , vα2 , α ∈( 0,1] 1
(
)
d H1 v1 , v 2 = ∫0 d H vα1 , vα2 dα . ∞ = v F : d= Denote supα >0 vα H ( v, I 0 )
(
)
K
, where I 0 is the fuzzy set taking value one at 0 and zero for all
x ≠ 0 . The space Fk ( X ) , d H∞ is a complete metric space (cf. [18], or [14]: Theorem 5.1.6) but not separable (cf. [17], or [14]: Remark 5.1.7). It is well known that vα = β 0 , there exists a finite 0 = α 0 < α1 < < α M = 1 , such that
(
)
d H ν α j ,ν α j −1 + ≤ ε , for all 1, , M .
Then for α j −1 < α < α j ,
1 n 1 n 1 n d H ∑ν αk ,ν α ≤ d H ∑ν αk j ,ν α j −1+ + d H ∑ν αk j −1+ ,ν α j = n k 1= n k 1 = n k 1 n n 1 1 ≤ d H ∑ν αk j ,ν α j + d H ∑ν αk j −1+ ,ν α j −1+ + 2d H ν α j ,ν α j −1+ n n = k 1= k1
(
)
Consequently,
1 n 1 n 1 n sup d H ∑ν αk ,ν α ≤ max d H ∑ν αk j ,ν α j + max d H ∑ν αk j −1+ ,ν α j −1+ + 2ε . α∈( 0,1] = n k 1= 1≤ j ≤ M n k 1 = 1≤ j ≤ M n k 1 Since the first two terms on the right hand converge to 0 in probability one, we have
1 n lim sup sup d H ∑ν αk ,ν α ≤ 2ε , n→∞ α ∈( 0,1] n k =1 but ε is arbitrary and the result follows. Theorem 3.6. Let X be a Banach space which satisfies the condition of Gα , let X n : n ≥ 1 dent fuzzy set-valued random variables in Fk ( X ) , such that E X n = I 0 for any n. If
{
∑ E φ0 ( ∞
X
j =1
j F
}
be indepen-
) < +∞,
where φ0 ( t ) = t1+α for 0 ≤ t ≤ 1 and φ0 ( t ) = t for t ≥ 1, then
∞
∑X j
converges with probability 1 in the sense
j =1
of d H∞ . Proof. Define j = U j X= I j , W j X jI
{X
F
}
≤1
{
}
j Note that X= W j + U j for each j, and both W j : j ≥ 1
fuzzy set-valued random variables. When
X
j F
{X
and
> 1 , we have W
j F
}
>1
{U
j F
. j
}
: j ≥1
= X
j F
are independent sequence of
, and φ0
( X )= j
F
X
j F
. Then,
for any m, n m m m E ∑W j ≤ ∑ E W j = ∑ E φ0 X j F . F = =j n jn j n= F
(
∞
And from
∑ E φ0 j =1
X
j
< ∞ , we know that E F
m
∑W j j =1
587
)
: m ≥ 1 is a Cauchy sequence. So, we have F
L. M. Shen, L. Guan
E
m
∑W j j =1
converges as m → ∞. F
Since convergence in the mean implied convergence in probability, Ito and Nisios result in [9] for independent random elements (cf. Section 4.5) provides that m
∑W j j =1
converges in probability 1 . F
So, for any n, m ≥ 1, m > n, by triangle inequality we have m n n m n j j j j j ∞ d H∞= ∑W , ∑W dH ∑W , ∑W + ∑ W = j 1 =j 1 = j 1 =j 1 =j n +1 n n m ≤ d H∞ ∑ W j , ∑ W j + d H∞ I 0 ∑ W j j= 1 j= 1 j= n +1 m = d H∞ I 0 , ∑ W j j= n +1
=
m
∑Wj
j = n +1
→ 0, a.e. as n, m → ∞. F
n It means ∑ W j : n ≥ 1 is a Cauchy sequence in the sense of d H∞ . By the completeness of j =1 n
we have
∑W j j =1
(F (X) , d ) , k
∞ H
converges almost everywhere in the sense of d H∞ . n
Next we shall prove that
∑U j j =1
converges in the sense of d H∞ . Firstly, we assume that
{U } j
fuzzy set-valued random variables. Then by the equivalent definition of Hausdorff metric, we have 1+α m m = E sup ∑ U βj E ∑U j j n= β ∈( 0,1] j n = F
1+α
K
m = E sup d H1+α ∑ U βj , {0} j =n β ∈( 0,1] = E sup sup β ∈( 0,1] x∗∈S ∗
m s x∗ , ∑U βj j =n
1+α
For any fixed n, m, there exists a sequence xk∗ ∈ S ∗ , such that m m lim s xk∗ , ∑ U βj = sup s x∗ , ∑ U βj . k →∞ = j n= j n x∗∈S ∗
That means there exist a sequence xk∗ ∈ S ∗ , such that 1+α 1+α m ∗ m j j = E sup lim s xk , ∑ U β . E ∑U β ∈( 0,1] k →∞ j n= = j n F
Then by Cr inequality, dominated convergence theorem and Lemma 3.2, we have
588
are all convex
E
L. M. Shen, L. Guan
1+α
1+α
m = E sup lim s xk∗ , ∑ U βj ∑U k →∞ j n= j n F β ∈( 0,1] m
j
1+α
m ≤ E sup lim ∑ s xk∗ , U βj β ∈( 0,1] k →∞ j = n
(
)
1+α
m ≤ E lim sup ∑ s xk∗ , U βj k →∞ β ∈( 0,1] j = n
(
)
1+α
m ≤ lim E sup ∑ s xk∗ , U βj − E s xk∗ , U βj + E s xk∗ , U βj k →∞ β ∈ 0,1 ( ] j =n
(
)
(
)
(
)
1+α
m = lim E lim ∑ s xk∗ , U βji − E s xk∗ , U βji + E s xk∗ , U βji k →∞ i →∞ j =n
(
)
(
)
(
)
1+α
m = lim lim E ∑ s xk∗ , U βji − E s xk∗ , U βji + E s xk∗ , U βji k →∞ i →∞ j =n
(
)
(
)
(
)
1+α
m m ≤ lim lim E ∑ s xk∗ , U βji − E s xk∗ , U βji + ∑ E s xk∗ , U βji k →∞ i →∞ = j n=j n
(
)
(
)
(
)
1+α 1+α m m ∗ j ∗ j ∗ j ≤ lim lim 2 E ∑ s xk , U βi − E s xk , U βi + ∑ E s xk , U β k →∞ i →∞ = j n j n=
(
1+α
)
(
)
(
)
1+α m 1+α m ∗ j ∗ j ∗ j E s xk , U βi ≤ lim lim 2 A∑ E s xk , U βi − E s xk , U βi + ∑ k →∞ i →∞ = j n j n=
(
1+α
)
(
)
(
)
1+α m m 1+α 1+α m 1+α j 1+α j j ∗ ∗ ∗ ≤ lim lim 2 2 A∑ E s xk , U βi + 2 A E s x , U + E s x , W ∑ ∑ k βi k βi k →∞ i →∞ = j n j n= = j n
(
1+α
)
(
)
(
)
1+α m m 1+α 1+α m 1+α ∗ j 1+α ∗ j ∗ j 2 , ≤ lim lim 2 2 A∑ E s xk , U βi + A E s x U + E s x , W ∑ ∑ k βi k βi k →∞ i →∞ = = j n j n= j n
(
1+α
)
(
)
1+α m 1+α m 2 +α j j ∗ ∗ lim lim 2 2 A∑ E s xk , U βi E s x , W + ∑ k βi k →∞ i →∞ = j n= j n
(
1+α
)
(
)
1+α m 1+α m 2 +α j j ∗ ∗ E sup s x , W ≤ lim 2 2 A∑ E sup s x , U βi + ∑ βi i →∞ j n= = x*∈S * j n x*∈S *
(
1+α
(
)
1+α
m m 1+α + E W j = lim 2 22 +α A∑ E U βji ∑ K i →∞ j n βi = j n= m m 1+α + E W j = 2 2 2 +α A ∑ E U j ∑ F j n j n= = 1+α
m m ≤ 2 22 +α A∑ E φ0 X j + ∑ E φ0 F = j n= j n 1+α
(
)
for each n and m.
589
K
1+α
1+α
F
(X ) j
F
1+α
)
(
)
L. M. Shen, L. Guan
Then, we know E
m
∑U j =1
1+α j F
is a Cauchy sequence. Hence, ∞
Thus by the similar way as above to prove
∑W j
{E ∑
m
Uj
j =1
F
} is a Cauchy sequence.
converges with probability 1 in the sense of d H∞ . We also
j =1
can prove that ∞
∑U j
converges
j =1
with probability 1 in the sense of d H∞ . In fact, for each n ≤ m , n n m n j m j ∞ j j j d H∞= ∑U , ∑U dH ∑U , ∑U + ∑ U = j 1 =j 1 = j 1 =j 1 =j n +1
≤
m
∑U j
j= n +1
F
→ 0, a.e.
as n, m → ∞.
So, we can prove that ∞
∑X j
converges
j =1
{ }
n
with probability 1 in the sense of d H∞ . If U j
∑ coU j
are not convex, we can prove
converges with proba-
j =1
bility 1 in the sense of d H∞ as above, and by the Lemma 3.5, we can prove that
n
∑U j
converges with proba-
j =1
bility 1 in the sense of d H∞ . Then the result was proved. From Theorem 3.6, we can easily obtain the following corollary. Corollary 3.7. Let X be a separable Banach space which is Gα for some 0 < α ≤ 1 . Let X n : n ≥ 1 be a sequence of independent fuzzy set-valued random variables in Fk ( X ) , such that E X n = I 0 for each n. If
{
1, 2, , are continuous and such that φn : + → + , n =
α n ⊂ + the convergence of ∞
∑
E φn
φn ( t ) t1+α and are non-decreasing, then for each t φn ( t )
( X ) n
φn (α n )
n =1
F
implies that ∞
Xn
∑α n =1
n
∞ H
converges with probability one in the sense of d . Proof. Let Xj Xj and W j Uj = I j I =
αj
If
Xn
F
{X
F
≤α j
> α n , by the non-decreasing property of
}
αj
φn ( t ) , we have t
(
n φn (α n ) φn X ≤ αn Xn
F
590
}
F
).
{X
j F
>α j
}
.
L. M. Shen, L. Guan
That is
Xn
αn Xn
If
F
F
≤
(
φn X n
F
φn (α n )
).
(4.1)
t1+α , we have φn ( t )
≤ α n , by the non-decreasing property of Xn
(
1+α F
φn X n
F
)
≤
α n1+α . φn (α n )
That is
Xn
α
1+α F
1+α n
≤
(
φn X n
F
φn (α n )
).
(4.2) ∞
∞
Then as the similar proof of Theorem 3.6, we can prove both
∑U j j =1
and
∑W j
converges with probability
j =1
one in the sense of d H∞ , and the result was obtained.
Acknowledgements We thank the Editor and the referee for their comments. Research of Li Guan is funded by the NSFC (11301015, 11571024, 11401016).
References [1]
Taylor, R.L. (1978) Lecture Notes in Mathematics. Springer-Verlag, 672.
[2]
Li, G. (2015) A Strong Law of Large Numbers for Set-Valued Random Variables in Gα Space. Journal of Applied Mathematics and Physics, 3, 797-801. http://dx.doi.org/10.4236/jamp.2015.37097
[3]
Artstein, Z. and Vitale, R.A. (1975) A Strong Law of Large Numbers for Random Compact Sets. Annals of Probability, 3, 879-882. http://dx.doi.org/10.1214/aop/1176996275
[4]
Hiai, F. (1984) Strong Laws of Large Numbers for Multivalued Random Variables, Multifunctions and Integrands. In: Salinetti, G., Ed., Lecture Notes in Mathematics, Vol. 1091, Springer, Berlin, 160-172.
[5]
Taylor, R.L. and Inoue, H. (1985) A Strong Law of Large Numbers for Random Sets in Banach Spaces. Bulletin of the Institute of Mathematics Academia Sinica, 13, 403-409.
[6]
Colubi, A., López-Díaz, M., Domnguez-Menchero, J.S. and Gil, M.A. (1999) A Generalized Strong Law of Large Numbers. Probability Theory and Related Fields, 114, 401-417. http://dx.doi.org/10.1007/s004400050229
[7]
Feng, Y. (2004) Strong Law of Large Numbers for Stationary Sequences of Random Upper Semicontinuous Functions. Stochastic Analysis and Applications, 22, 1067-1083. http://dx.doi.org/10.1081/SAP-120037631
[8]
Molchanov, I. (1999) On Strong Laws of Large Numbers for Random Upper Semicontinuous Functions. Journal of Mathematical Analysis and Applications, 235, 249-355. http://dx.doi.org/10.1006/jmaa.1999.6403
[9]
Puri, M.L. and Ralescu, D.A. (1991) Convergence Theorem for Fuzzy Martingales. Journal of Mathematical Analysis and Applications, 160, 107-121. http://dx.doi.org/10.1016/0022-247X(91)90293-9
[10] Li, S. and Ogura, Y. (2003) A Convergence Theorem of Fuzzy Valued Martingale in the Extended Hausdorff Metric H∞. Fuzzy Sets and Systems, 135, 391-399. http://dx.doi.org/10.1016/S0165-0114(02)00145-8 [11] Li, S. and Ogura, Y. (2003) Strong Laws of Numbers for Independent Fuzzy Set-Valued Random Variables. Fuzzy Sets and Systems, 157, 2569-2578. http://dx.doi.org/10.1016/j.fss.2003.06.011 [12] Inoue, H. (1991) A Strong Law of Large Numbers for Fuzzy Random Sets. Fuzzy Sets and Systems, 41, 285-291. http://dx.doi.org/10.1016/0165-0114(91)90132-A [13] Guan, L. and Li, S. (2004) Laws of Large Numbers for Weighted Sums of Fuzzy Set-Valued Random Variables. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12, 811-825.
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http://dx.doi.org/10.1142/S0218488504003223
[14] Li, S., Ogura, Y. and Kreinovich, V. (2002) Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables. Kluwer Academic Publishers, Dordrecht. http://dx.doi.org/10.1007/978-94-015-9932-0 [15] Beer, G. (1993) Topologies on Closed and Closed Convex Sets. Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, Holland. http://dx.doi.org/10.1007/978-94-015-8149-3 [16] Chen, Y. (1984) Fuzzy Systems and Mathematics. Huazhong Institute Press of Science and Technology, Wuhan. (In Chinese) [17] Klement, E.P., Puri, L.M. and Ralescu, D.A. (1986) Limit Theorems for Fuzzy Random Variables. Proceedings of the Royal Society of London A, 407, 171-182. http://dx.doi.org/10.1098/rspa.1986.0091 [18] Puri, M.L. and Ralescu, D.A. (1986) Fuzzy Random Variables. Journal of Mathematical Analysis and Applications, 114, 406-422. http://dx.doi.org/10.1016/0022-247X(86)90093-4 [19] Li, S. and Ogura, Y. (1996) Fuzzy Random Variables, Conditional Expectations and Fuzzy Martingales. Journal of Fuzzy Mathematics, 4, 905-927.
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