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Strong convergence for weighted sums of fuzzy random sets
Information Sciences 176 (2006) 1086–1099 www.elsevier.com/locate/ins
Strong convergence for weighted sums of fuzzy random sets q Sang Yeol Joo a,1, Yun Kyong Kim Joong Sung Kwon c,2
b,*
,
a
Department of Statistics, Kangwon National University, Chuncheon, Kangwon 200-701, South Korea b Department of Information and Communication Engineering, Dongshin University, Naju, Chonnam 520-714, South Korea c Department of Mathematics, Sunmoon University, Asan, Chungnam 336-708, South Korea Received 10 June 2004; accepted 14 February 2005
Abstract In this paper, we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of uppersemicontinuous fuzzy sets in Rp. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Random sets; Fuzzy random sets; Strong law of large numbers; Uniform integrability; Tightness; Weighted sums
q
This paper was supported by Korea Research Foundation Grant (KRF-2002-070-C00013). Corresponding author. Tel.: +82 61 330 3312. E-mail addresses: [email protected] (S.Y. Joo), [email protected] (Y.K. Kim), jskwon@ omega.sunmoon.ac.kr (J.S. Kwon). 1 Tel.: +82 33 250 8433. 2 Tel.: +82 41 530 2222. *
0020-0255/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2005.02.002
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1. Introduction In recent years, strong laws of large number for independent fuzzy random sets can be found in the literature (e.g. [2,3,6,8,10,13,15,21]). These results are generalizations of SLLN for random sets established by Artstein and Vitale [1], Taylor and Inoue [19], Uemura [22], and so on. However, strong convergence for weighted sums of independent fuzzy random sets has not been established until now. P Pruitt [16] obtained almost sure convergence for weighted sums nk¼1 ank X k of independent identically distributed real-valued random variables {Xn} by assuming 1þ1c
EjX 1 j
<1
and
max jank j ¼ Oðnc Þ for some c > 0:
16k6n
Rohatgi [18] extended PruittÕs result to independent, but not necessarily identically distributed random variables {Xn} by requiring stochastically 1þ1 bounded condition, i.e., there exists a random variable X with EjX j c < 1 such that for each n, P ðjX n j P kÞ 6 P ðjX j P kÞ for all k > 0: Taylor and Inoue [20] showed that the similar results can be obtained for independent random sets. The purpose of this paper is to generalize the results of Taylor and Inoue [20] to the case of fuzzy random sets. This will be proceeded by considering uniformly integrable fuzzy random sets taking values in the space of upper-semicontinuous fuzzy sets in Rp.
2. Preliminaries Let K ¼ KðRp Þ denote the family of non-empty compact subsets of the Euclidean space Rp. Then the space K is metrizable by the Hausdorff metric h defined by hðA; BÞ ¼ max sup inf ja bj; sup inf ja bj : a2A b2B
b2B a2A
A norm of A 2 K is defined by kAk ¼ hðA; f0gÞ ¼ sup jaj: a2A
It is well-known that K is complete and separable with respect to the Hausdorff metric h (see [5]).
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The addition and scalar multiplication on K are defined as usual: A B ¼ fa þ b : a 2 A; b 2 Bg; kA ¼ fka : a 2 Ag for A; B 2 K and k 2 R. Let F ¼ FðRp Þ denote the family of all fuzzy sets e u : Rp ! ½0; 1 with the following properties; (1) e u is normal, i.e., there exists x 2 Rp such that e u ðxÞ ¼ 1; (2) e u is upper semicontinuous; (3) supp e u ¼ clfx 2 Rp : e u ðxÞ > 0g is compact, where cl denotes the closure. For a fuzzy subset e u of Rp, the a-level set of e u is defined by fx : e u ðxÞ P ag; if 0 < a 6 1; La e u¼ supp e u; if a ¼ 0: Then, it follows immediately that e u2F
if and only if La e u2K
for each a 2 ½0; 1:
u ðxÞ > ag by Laþ e u , then Also, if we denote clfx 2 Rp : e u ; Laþ e u Þ ¼ 0: lim hðLb e b#a
The linear structure on F is also defined as usual; u ðxÞ; ev ðyÞÞ; ðe u ev ÞðzÞ ¼ sup minðe xþy¼z
( ðke u ÞðzÞ ¼
e u ðz=kÞ; k 6¼ 0; e 0ðzÞ; k¼0
for e u ; ev 2 F and k 2 R, where e 0 ¼ I f0g denotes the indicator function of {0}. Then it is known that for each a 2 [0, 1], u ev Þ ¼ La e u La ev ; La ðe
u Þ ¼ kLa e u: La ðke
Recall that a fuzzy subset e u of Rp is said to be convex if e u ðkx þ ð1 kÞyÞ P minðe u ðxÞ; e u ðyÞÞ for x, y 2 Rp and k 2 [0, 1]. The convex hull of e u is defined by coðe u Þ ¼ inffev : ev is convex and ev P e u g: u Þ ¼ coðLa e u Þ (for details, see [14]). Then for each a 2 [0, 1], La coðe Now we define the uniform metric d1 on F as usual; u ; ev Þ ¼ sup hðLa e u ; La ev Þ: d 1 ðe 06a61
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Also, the norm of e u is defined as u; e 0Þ ¼ kL0 e u k ¼ sup jxj: ke u k ¼ d 1 ðe x2L0 ~ u
It is well-known that ðF; d 1 Þ is complete but is not separable (see [13]). However, we can introduce the Skorokhod metric ds on F which makes it separable and topologically complete as follows (for details, see [9]). Definition 2.1. Let T denote the class of strictly increasing, continuous mapping of [0, 1] onto itself. For e u ; ev 2 F, we define d s ðe u ; ev Þ ¼ inf > 0 : there exists a t 2 T such that sup jtðaÞ aj 6 06a61 and d 1 ðe u ; tðev ÞÞ 6 ; where tðev Þ denotes the composition of ev and t.
3. Main results Throughout this paper, let ðX; A; P Þ be a probability space. A set-valued function X : X ! ðK; hÞ is called a random set if it is measurable. A random set X is said to be integrably bounded if EkXk < 1. The expectation of integrably bounded random set X is defined by EðX Þ ¼ fEðnÞ : n 2 LðX; Rp Þ and nðxÞ 2 X ðxÞ a:s:g; where L(X, Rp) denotes the class of all Rp-valued random variables n such that Ejnj < 1 (see [7]). e : X ! F is called a fuzzy random set if for A fuzzy set valued function X e each a 2 [0, 1], La X is a random set. This definition was introduced by Puri and Ralescu [17] as a natural generalization of a random set. There they [17] used a term ‘‘fuzzy random variable’’ instead of ‘‘fuzzy random set’’. It is e is a fuzzy random set if and only if X e : X ! ðF; d s Þ is meawell-known that X surable (for details, see [4,11]). e is said to be integrably bounded if Ek X e k < 1. The A fuzzy random set X e eÞ expectation of integrably bounded fuzzy random set X is a fuzzy subset Eð X p of R defined by, e ÞðxÞ ¼ supfa 2 ½0; 1 : x 2 EðLa X e Þg: Eð X e Þ (see [17]). e Þ ¼ EðLa X It is well-known that for each a 2 [0, 1], La Eð X e n g be a sequence of integrably bounded fuzzy random sets and {kni} Let f X be a Toeplitz sequence, i.e., {kni} is a double array of real numbers satisfying
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(1) For each i, limn!1 kni = 0. P 1 (2) There exists C > 0 such that i¼1 jkni j 6 C for each n. The problem which we will consider is to find sufficient conditions for strong e n g in the following; convergence of weighted sums of f X n n e e lim d 1 kni X i ; kni coðE X i Þ ¼ 0 a:s:; n!1
i¼1
i¼1
e i Þ denotes the convex hull of Eð X e i Þ. where coðE X To this end, we require the following assumption. For each > 0; there exists a partition 0 = a0 < a1 < < am = 1 of [0, 1] such that for all n, e n ; Lak X e n Þ < : max EhðLaþk1 X
16k6m
ð3:1Þ
e n g is identically distributed, then it satThe next theorem implies that if f X isfies the condition (3.1). e 1 k < 1, Then for each > 0; there exists a partition Theorem 3.1. (a) Let Ek X 0 = a0 < a1 < < am = 1 of [0, 1] such that e ; Lak X e Þ < : max EhðLaþ X k1
16k6m
e n g is identically distributed and Ek X e 1 k < 1, then it satisfies the condi(b) If f X tion (3.1). Proof. (a) Suppose that the assertion does not hold. Then we can find a 0 > 0 such that for any partition P : 0 ¼ a0 < a1 < < am ¼ 1 of [0, 1], there is a ak in P satisfying e ; Lak X e Þ P 0 : EhðLaþ X k1
We consider a sequence fPn g of partitions of [0, 1] such that (1) Pnþ1 is a refinement of Pn ; (2) kPn k ! 0, where kPn k denotes the longest length of largest subinterval of kPn k. For each n, let akn be the largest point of Pn such that e ; Lak X e Þ P 0 : EhðLaþ X k n 1
n
ð3:2Þ
Then fakn g is monotone decreasing since Pnþ1 is a refinement of Pn . By (3.2), we can choose a sequence {bn} with akn 1 < bn < akn such that for all n, e ; Lak X e Þ > 0 =2: EhðLbn X n
ð3:3Þ
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Since fakn g is monotone decreasing and kPn k ! 0, it follows that e ; Lak X e Þ ! 0 pointwise: hðLb X n
n
Then by the Lebesgue dominated convergence theorem, e ; Lak X e Þ ! 0; EhðLb X n
n
which is impossible by (3.3). (b) It follows from the fact that e n ; Lak X e n Þ ¼ EhðLaþ X e 1 ; Lak X e 1 Þ: EhðLaþk1 X k1
Now in order to obtain strong convergence of weighted sums of independent fuzzy random sets, we need the concept of independence for fuzzy random sets. By the way, there are several definitions of independence for fuzzy random sets. In this paper, we adopt the following notion of independence. e n g be a sequence of fuzzy random sets. Then f X e n g is Definition 3.2. Let f X e n Þg is independent, where rð X e Þ is the called independent if the r-fields frð X e : X ! ðF; d s Þ is measurable. smallest r-field of subsets of X such that X e j0 6 a 6 1gÞ because X e : X ! ðF; d s Þ is measure Þ ¼ rðfLa X Note that rð X e : X ! ðK; hÞ is measurable. able if and only if for each a 2 [0, 1], La X e n g be a sequence of independent fuzzy random sets Theorem 3.3. Let f X satisfying (3.1). Suppose that there exists a nonnegative random variable X with 1 EX 1þc < 1 for some c > 0 such that for each n, e n k P kÞ 6 P ðX P kÞ P ðk X
for all k > 0:
ð3:4Þ
If {kni} is a Toeplitz sequence satisfying max16i6njknij = O(nc), then n n e e lim d 1 kni X i ; kni coðE X i Þ ¼ 0 a:s: n!1
i¼1
i¼1
Proof. Since {kni} is a Toeplitz sequence, there exist a positive number C such that 1 X jkni j 6 C for all n: i¼1
Let > 0 be given. By assumption (3.1), we choose 0 = a0 < a1 < < am = 1 of [0, 1] such that for all n e n ; Lak X e n Þ < =C: max EhðLaþk1 X
16k6m
ð3:5Þ
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Then
S.Y. Joo et al. / Information Sciences 176 (2006) 1086–1099
n n e i ; kni La coðE X e iÞ sup h kni La X i¼1
ak1
i¼1
n n n n e e e e sup h kni La X i ; kni Lak X i þ h kni Lak X i ; kni Lak coðE X i Þ
6
i¼1
ak1
i¼1
n
n
i¼1
i¼1
i¼1
e i Þ; kni La coðE X e iÞ þh kni Lak coðE X 6
n X
n n e e e e jkni jhðLaþk1 X i ;Lak X i Þ þ h kni Lak X i ; kni Lak coðE X i Þ i¼1
i¼1
þ
i¼1
n X
i¼1
e i Þ;Lak coðE X e i ÞÞ ¼ ðIÞ þ ðIIÞ þ ðIIIÞ: jkni jhðLaþk1 coðE X
i¼1
For (I), we first note that for all n, e n ; Lak X e n Þ P kg 6 P f2k X e n k P kg 6 P ð2X P kÞ: P fhðLaþk1 X By RohatgiÕs result [18] for real-valued random variables, n X
e i ; Lak X e i Þ EhðLaþ X e i ; Lak X e i Þ ! 0 a:s: jkni j½hðLaþk1 X k1
i¼1
Thus by (3.5), n X e i ; Lak X e i Þ EhðLaþ X e i ; Lak X e i Þ ðIÞ ¼ jkni j½hðLaþk1 X k1 i¼1
þ
n X
e i ; Lak X e i Þ < a:s: for large n: jkni jEhðLaþk1 X
i¼1
e n kg is uniformly For (II), first we note that assumption (3.4) implies that fk X integrable. For, Z Z 1 e n k dP ¼ aP ðk X e n k > aÞ þ e n k > xÞ dx kX P ðk X fke X n k>ag a Z 1 Z 6 aP ðX > aÞ þ P ðX > xÞ dx ¼ X dP a
fX >ag
e n kg. Then it follows that for each a, implies the uniform integrability of fk X e fLa ; X n g is uniformly integrable. Recall that uniform integrability is equivalent to compactly uniform integrablity for a sequence of Rp-valued random sets. Thus we conclude from Theorem 3.3 of Taylor and Inoue [20] for random sets that ðIIÞ ! 0 a:s:
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Finally, for (III) by (3.5) and convexity inequality, we have n n X X e i ; co Lak X e iÞ 6 e i ; Lak X e i Þ < : jkni jEhðco Laþk1 X jkni jEhðLaþk1 X ðIIIÞ ¼ i¼1
i¼1
Therefore, n n e e d 1 kni X i ; kni coðE X i Þ i¼1 i¼1 n n e e ¼ max sup h kni La X i ; La kni coðE X i Þ < 2 a:s: for large n: 16k6m ak 1
i¼1
i¼1
Since is arbitrary, this completes the proof.
h
The following Corollary is an analogue of strong law of large numbers obtained by Mochanov [15]. e n g be a sequence of independent and identically distributed Corollary 3.4. Let f X 1 e 1 k1þc < 1 for some c > 0. If {kni} is a Toeplitz fuzzy random sets with Ek X sequence satisfying max16i6njknij = O(nc), then ! ! n X n e e lim d 1 kni X i ; kni coðE X 1 Þ ¼ 0 a:s: n!1
i¼1
i¼1
e n g be a sequence of independent fuzzy random sets satisfyCorollary 3.5. Let f X ing (3.1). If e n kr ¼ M < 1 sup Ek X
for some r > 1
ð3:6Þ
n
then for any Toeplitz sequence {kni} satisfying max16i6n jkni j ¼ Oðnc Þ for 1 c > r1 , n n e i ; kni coðE X e i Þ ¼ 0 a:s: lim d 1 kni X n!1
i¼1
i¼1
Proof. By Lemma 3 of Wei and Taylor [23], (3.6) implies that there exists a non1 negative random variable X with EX 1þc < 1 for 0 < 1c < r 1 such that (3.4) is satisfied for each n. Thus the result follows immediately from Theorem 3.3. h e n 2 Kg ¼ 1 Remark. If there exists a compact subset K of ðF; d s Þ such that P f X for all n, then (3.6) is also satisfied for all r > 1. Recall that we can define the concept of convexity on F as in the case of a vector space. That is, A F is said to be convex if ke u ð1 kÞev 2 A whenever
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e u , ev 2 A and 0 6 k 6 1. Using this concept of convexity on F, Kim [12] obtained the following; Theorem 3.6. Let us denote by CðFÞ the collection of all relatively compact subsets K of ðF; d s Þ for which the convex hull of K is also relatively compact. Then K 2 CðFÞ if and only if the following two conditions hold: (1) supfke uk : e u 2 Kg < 1. (2) For each > 0, there exists a finite partition 0 = a0 < a1 < < am = 1 of [0, 1], such that u; Lak e u Þ 6 : sup max hðLaþk1 ~ eu2K 16k6m By applying this result, we can obtain the following. e n g be a sequence of independent and convexly tight fuzzy Corollary 3.7. Let f X random sets, i.e., for each > 0, there exists a compact convex subset K of ðF; d s Þ such that e n 62 KÞ < for all n: P ðX e n k ¼ M < 1 for some r > 1, then for any Toeplitz sequence {kni} If supn Ek X 1 satisfying max16i6njknij = O(nc) for c > r1 , n n e i ; kni coðE X e i Þ ¼ 0 a:s: lim d 1 kni X r
n!1
i¼1
i¼1
e n g satisfies the assumption (3.1) in view of Proof. It suffices to prove that f X e n g, we can choose Corollary 3.5. Let > 0 be given. By convex tightness of f X a compact convex subset K of ðF; d s Þ such that e n 62 KÞ < ð=4Þr=ðr1Þ M 1=ðr1Þ P ðX
for all n:
Then e n kÞ 6 ðEk X e n kr Þ1=r P ð X e n 62 KÞðr1Þ=r EðI fe kX X n 62Kg
e n 62 2KÞðr1Þ=r < =4: 6 M 1=r P ð X
Now by Theorem 3.6, there exists a finite partition 0 = a0 < a1 < < am = 1 of [0, 1], such that sup max hðLaþk1 ~ u; Lak ~ uÞ 6 =2: eu2K 16k6m
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Thus, we have for all n, e n ; Lak X e n Þ 6 max EI e n ; Lak X e nÞ hðLaþk1 X max EhðLaþk1 X fe X n 2Kg 16k6m
16k6m
e n ; Lak X e nÞ þ max EI fe hðLaþk1 X X n 62Kg 16k6m e n k < : 6 =2 þ 2EI fe kX X n 62Kg h 1=n; if 1 6 i 6 n; then we can obtain a If we apply Theorem 3.3 to kni ¼ 0; if i > n; SLLN for independent fuzzy random sets. But in this case, we need the restrictive condition EjX j2 < 1. Similarly, we need the restrictive condition e n kr < 1 for some r > 2 in order to provide a SLLN by applying Corsupn Ek X ollaries 3.5 and 3.7. However, we can obtain much better SLLN by similar arguments as in the proof of Theorem 3.3. This yields the desired result.
e n g be a sequence of independent fuzzy random sets Theorem 3.8. Let f X satisfying (3.1). Suppose that the following two conditions are satisfied; e n kg is uniformly integrable. (1) fk X P1 1 e n kr < 1 for some 1 6 r 6 2. (2) n¼1 nr Ek X Then
n n 1 e i ; coðE X e i Þ ¼ 0 a:s: d1 X n i¼1 i¼1
lim
n!1
Proof. Let > 0 be given. By assumption (3.1), we choose 0 = a0 < a1 < < am = 1 of [0, 1] such that for all n, e n ; Lak X e n Þ < : max16k6m EhðLaþk1 X
ð3:7Þ
Now, 1 n
sup ak1
6
1 n
n n e i ; La coðE X e iÞ h La X i¼1
sup ak1
i¼1
n n n n e e e e h La X i ; Lak X i þ h Lak X i ; i¼1 Lak coðE X i Þ i¼1
i¼1
i¼1
n n e i Þ; La coðE X e iÞ þh Lak coðE X i¼1
i¼1
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n n n 1X 1 e e e e 6 hðLaþk1 X i ; Lak X i Þ þ h Lak X i ; Lak coðE X i Þ n i¼1 n i¼1 i¼1 n 1X e i Þ; Lak coðE X e i ÞÞ ¼ ðIÞ þ ðIIÞ þ ðIIIÞ: þ hðLaþk1 coðE X n i¼1 For (I), we first note that for all n, e n ; Lak X e n Þ 6 2r Ek X e nk : E½hðLaþk1 X r
r
By ChungÕs SLLN for real-valued random variables, n X 1 e i ; Lak X e i Þ EhðLaþ X e i ; Lak X e i Þ ! 0 a:s: ½hðLaþk1 X k1 n i¼1
Thus, by (3.7), ðIÞ ¼
n 1X e i ; Lak X e i Þ EhðLaþ X e i ; Lak X e i Þ ½hðLaþk1 X k1 n i¼1 n 1X e i ; Lak X e i Þ < a:s: for large n: þ EhðLaþk1 X n i¼1
e n g is compactly uniformly integrable, it follows from For (II), since fLak X SLLN for random sets (see [20]) that ðIIÞ ! 0 a:s: Finally, for (III) by (3.7), we have ðIIIÞ 6
n 1X e i ; La X e i Þ 6 : EhðLaþk1 X n i¼1
Therefore, n n 1 e i ; coðE X e i Þ ¼ 1 max d1 X n n 16k6m i¼1 i¼1
sup ak1
n n e i ; La coðE X e iÞ h La X i¼1
i¼1
6 2 a:s: for large n: Since is arbitrary, this completes the proof.
h
e n g be a sequence of independent fuzzy random sets satisfyCorollary 3.9. Let f X e n kg is tight ing (3.1). Suppose that a sequence of real-valued random variables fk X and e n k < 1 for some r > 1: sup Ek X r
n
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Then
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n n 1 e e lim d 1 X i ; coðE X i Þ ¼ 0 a:s: n!1 n i¼1 i¼1
Proof. This follows from the fact that tightness and rth (r > 1) moments cone n kg imply uniform integrability of fk X e n kg. h dition of fk X e n g be a sequence of independent and convexly tight fuzzy Corollary 3.10. Let f X e n kr ¼ M < 1 for some r > 1, then random sets. If supn Ek X n n 1 e e lim d 1 X i ; coðE X i Þ ¼ 0 a:s: n!1 n i¼1 i¼1 e n g implies of tightness of fk X e n kg. Proof. It is trivial that convex tightness of f X e n g satisfies the assumption (3.1). h By the proof of Corollary 3.7, f X Finally, we will give an example in order to apply our results Example. Let e u 2 F be fixed. For each n, we let 1 e n ¼ n~ P ðX uÞ ¼ ; n
1 en ¼ ~ P ðX 0Þ ¼ 1 : n e e n ; Lb X e nÞ ¼ e Then it follows that Eð X n Þ ¼ e u and Ek X n k ¼ ke u k. Since EhðLa X e n g is not e n g satisfy (3.1). But f X u ; Lb e u Þ, it can be easily proved that f X hðLa e tight. For, if K is a compact subset of F, then there exists a n0 such that n0 e u 62 K. And so, Z e n k dP ¼ ke kX u k: sup n fe X n 62Kg Now let f Ye n g be independent fuzzy random sets such that for some r > 0, 1 P ð Ye n ¼ ne uÞ ¼ r ; n
1 P ð Ye n ¼ e 0Þ ¼ 1 r : n u k. Also, since u and Ek Ye n k ¼ n1r ke Then it can be shown that Eð Ye n Þ ¼ n1r e 1r EhðLa Ye n ; Lb Ye n Þ ¼ n hðLa e u ; Lb e u Þ, it satisfies the condition (3.1). Since u kr , by Corollary 3.5, Ek Ye n kr ¼ ke ! ! n X n k ni lim d 1 kni Ye i ; coðe u Þ ¼ 0 a:s: n!1 i¼1 ir1 i¼1 for any Toeplitz sequence {kni} satisfying max16i Also, since fk Ye n kg is tight, by Corollary 3.9,
6nkknik
1 = O(nc) for c > r1 .
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! ! n X n 1 1 lim d 1 Ye i ; coðe u Þ ¼ 0 a:s: n!1 n i¼1 ir1 i¼1
4. Concluding remarks In this study, we generalized the results of strong convergence for weighted sums of independent random sets presented by Taylor and Inoue [20]. It is still to be demonstrated whether similar results hold in the case assumption (3.1) has not been imposed. References [1] Z. Artstein, R.A. Vitale, A strong law of large numbers for random compact sets, Ann. Probab. 3 (1975) 879–882. [2] A. Colubi, J.S. Dominguez-Menchero, M. Lo´pez-Dia´z, M.A. Gil, A generalized strong law of large numbers, Probab. Theory Rel. Fields 114 (1999) 401–417. [3] A. Colubi, J.S. Dominguez-Menchero, M. Lo´pez-Dia´z, R. Ko¨rner, A method to derive strong laws of large numbers for random upper semicontinuous functions, Statist. Probab. Lett. 53 (2001) 269–275. [4] A. Colubi, J.S. Dominguez-Menchero, M. Lopez-Diaz, D. Ralescu, On the formalization of fuzzy random variables, Inform. Sci. 133 (2001) 3–6. [5] G. Debreu, Integration of correspondences, Proc. 5th Berkeley Symp. Math. Statist. Prob. 2 (1966) 351–372. [6] Y. Feng, An approach to generalize law of large numbers for fuzzy random variables, Fuzzy Sets Syst. 128 (2002) 237–245. [7] F. Hiai, H. Umegaki, Integral, conditional expectation and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149–182. [8] H. Inoue, A strong law of large numbers for fuzzy random sets, Fuzzy Sets Syst. 41 (1991) 285– 291. [9] S.Y. Joo, Y.K. Kim, Topological properties on the space of fuzzy sets, J. Math. Anal. Appl. 246 (2000) 576–590. [10] Y.K. Kim, Strong law of large numbers for random upper semicontinuous functions, Bull. Kor. Math. Soc. 39 (2002) 511–526. [11] Y.K. Kim, Measurability for fuzzy valued functions, Fuzzy Sets Syst. 129 (2002) 105–109. [12] Y.K. Kim, Compactness and convexity on the space of fuzzy sets II, Nonlinear Anal. Appl. 57 (2004) 639–653. [13] E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. Lond. Ser. A 407 (1986) 171–182. [14] R. Lowen, Convex fuzzy sets, Fuzzy Sets Syst. 3 (1980) 291–310. [15] I. Molchanov, On strong law of large numbers for random upper semicontinuous functions, J. Math. Anal. Appl. 235 (1999) 349–355. [16] W. Pruitt, Summability of independent random variables, J. Math. Mech. Appl. 15 (1966) 769– 776. [17] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409– 422. [18] V.K. Rohatgi, Convergence of weighted sums of independent random variables, Proc. Cambridge Philos. Soc. 69 (1971) 305–307.
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[19] R.L. Taylor, H. Inoue, A strong law of large numbers for random sets in Banach space, Bull. Inst. Math. Acad. Sin. 13 (1985) 403–409. [20] R.L. Taylor, H. Inoue, Convergence of weighted sums of random sets, Stoch. Anal. Appl. 3 (1985) 379–396. [21] R.L. Taylor, L. Seymour, Y. Chen, Strong laws of large numbers for fuzzy random sets, University of Georgia, Department of Statistics, Technical report 2000-7. [22] T. Uemura, A law of large numbers for random sets, Fuzzy Sets Syst. 59 (1993) 181–188. [23] D. Wei, R.L. Taylor, Convergence of weighted sums of tight random elements, J. Multivariate Anal. 8 (1978) 282–294.
Strong convergence for weighted sums of fuzzy random sets q Sang Yeol Joo a,1, Yun Kyong Kim Joong Sung Kwon c,2
b,*
,
a
Department of Statistics, Kangwon National University, Chuncheon, Kangwon 200-701, South Korea b Department of Information and Communication Engineering, Dongshin University, Naju, Chonnam 520-714, South Korea c Department of Mathematics, Sunmoon University, Asan, Chungnam 336-708, South Korea Received 10 June 2004; accepted 14 February 2005
Abstract In this paper, we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of uppersemicontinuous fuzzy sets in Rp. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Random sets; Fuzzy random sets; Strong law of large numbers; Uniform integrability; Tightness; Weighted sums
q
This paper was supported by Korea Research Foundation Grant (KRF-2002-070-C00013). Corresponding author. Tel.: +82 61 330 3312. E-mail addresses: [email protected] (S.Y. Joo), [email protected] (Y.K. Kim), jskwon@ omega.sunmoon.ac.kr (J.S. Kwon). 1 Tel.: +82 33 250 8433. 2 Tel.: +82 41 530 2222. *
0020-0255/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2005.02.002
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1. Introduction In recent years, strong laws of large number for independent fuzzy random sets can be found in the literature (e.g. [2,3,6,8,10,13,15,21]). These results are generalizations of SLLN for random sets established by Artstein and Vitale [1], Taylor and Inoue [19], Uemura [22], and so on. However, strong convergence for weighted sums of independent fuzzy random sets has not been established until now. P Pruitt [16] obtained almost sure convergence for weighted sums nk¼1 ank X k of independent identically distributed real-valued random variables {Xn} by assuming 1þ1c
EjX 1 j
<1
and
max jank j ¼ Oðnc Þ for some c > 0:
16k6n
Rohatgi [18] extended PruittÕs result to independent, but not necessarily identically distributed random variables {Xn} by requiring stochastically 1þ1 bounded condition, i.e., there exists a random variable X with EjX j c < 1 such that for each n, P ðjX n j P kÞ 6 P ðjX j P kÞ for all k > 0: Taylor and Inoue [20] showed that the similar results can be obtained for independent random sets. The purpose of this paper is to generalize the results of Taylor and Inoue [20] to the case of fuzzy random sets. This will be proceeded by considering uniformly integrable fuzzy random sets taking values in the space of upper-semicontinuous fuzzy sets in Rp.
2. Preliminaries Let K ¼ KðRp Þ denote the family of non-empty compact subsets of the Euclidean space Rp. Then the space K is metrizable by the Hausdorff metric h defined by hðA; BÞ ¼ max sup inf ja bj; sup inf ja bj : a2A b2B
b2B a2A
A norm of A 2 K is defined by kAk ¼ hðA; f0gÞ ¼ sup jaj: a2A
It is well-known that K is complete and separable with respect to the Hausdorff metric h (see [5]).
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The addition and scalar multiplication on K are defined as usual: A B ¼ fa þ b : a 2 A; b 2 Bg; kA ¼ fka : a 2 Ag for A; B 2 K and k 2 R. Let F ¼ FðRp Þ denote the family of all fuzzy sets e u : Rp ! ½0; 1 with the following properties; (1) e u is normal, i.e., there exists x 2 Rp such that e u ðxÞ ¼ 1; (2) e u is upper semicontinuous; (3) supp e u ¼ clfx 2 Rp : e u ðxÞ > 0g is compact, where cl denotes the closure. For a fuzzy subset e u of Rp, the a-level set of e u is defined by fx : e u ðxÞ P ag; if 0 < a 6 1; La e u¼ supp e u; if a ¼ 0: Then, it follows immediately that e u2F
if and only if La e u2K
for each a 2 ½0; 1:
u ðxÞ > ag by Laþ e u , then Also, if we denote clfx 2 Rp : e u ; Laþ e u Þ ¼ 0: lim hðLb e b#a
The linear structure on F is also defined as usual; u ðxÞ; ev ðyÞÞ; ðe u ev ÞðzÞ ¼ sup minðe xþy¼z
( ðke u ÞðzÞ ¼
e u ðz=kÞ; k 6¼ 0; e 0ðzÞ; k¼0
for e u ; ev 2 F and k 2 R, where e 0 ¼ I f0g denotes the indicator function of {0}. Then it is known that for each a 2 [0, 1], u ev Þ ¼ La e u La ev ; La ðe
u Þ ¼ kLa e u: La ðke
Recall that a fuzzy subset e u of Rp is said to be convex if e u ðkx þ ð1 kÞyÞ P minðe u ðxÞ; e u ðyÞÞ for x, y 2 Rp and k 2 [0, 1]. The convex hull of e u is defined by coðe u Þ ¼ inffev : ev is convex and ev P e u g: u Þ ¼ coðLa e u Þ (for details, see [14]). Then for each a 2 [0, 1], La coðe Now we define the uniform metric d1 on F as usual; u ; ev Þ ¼ sup hðLa e u ; La ev Þ: d 1 ðe 06a61
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Also, the norm of e u is defined as u; e 0Þ ¼ kL0 e u k ¼ sup jxj: ke u k ¼ d 1 ðe x2L0 ~ u
It is well-known that ðF; d 1 Þ is complete but is not separable (see [13]). However, we can introduce the Skorokhod metric ds on F which makes it separable and topologically complete as follows (for details, see [9]). Definition 2.1. Let T denote the class of strictly increasing, continuous mapping of [0, 1] onto itself. For e u ; ev 2 F, we define d s ðe u ; ev Þ ¼ inf > 0 : there exists a t 2 T such that sup jtðaÞ aj 6 06a61 and d 1 ðe u ; tðev ÞÞ 6 ; where tðev Þ denotes the composition of ev and t.
3. Main results Throughout this paper, let ðX; A; P Þ be a probability space. A set-valued function X : X ! ðK; hÞ is called a random set if it is measurable. A random set X is said to be integrably bounded if EkXk < 1. The expectation of integrably bounded random set X is defined by EðX Þ ¼ fEðnÞ : n 2 LðX; Rp Þ and nðxÞ 2 X ðxÞ a:s:g; where L(X, Rp) denotes the class of all Rp-valued random variables n such that Ejnj < 1 (see [7]). e : X ! F is called a fuzzy random set if for A fuzzy set valued function X e each a 2 [0, 1], La X is a random set. This definition was introduced by Puri and Ralescu [17] as a natural generalization of a random set. There they [17] used a term ‘‘fuzzy random variable’’ instead of ‘‘fuzzy random set’’. It is e is a fuzzy random set if and only if X e : X ! ðF; d s Þ is meawell-known that X surable (for details, see [4,11]). e is said to be integrably bounded if Ek X e k < 1. The A fuzzy random set X e eÞ expectation of integrably bounded fuzzy random set X is a fuzzy subset Eð X p of R defined by, e ÞðxÞ ¼ supfa 2 ½0; 1 : x 2 EðLa X e Þg: Eð X e Þ (see [17]). e Þ ¼ EðLa X It is well-known that for each a 2 [0, 1], La Eð X e n g be a sequence of integrably bounded fuzzy random sets and {kni} Let f X be a Toeplitz sequence, i.e., {kni} is a double array of real numbers satisfying
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(1) For each i, limn!1 kni = 0. P 1 (2) There exists C > 0 such that i¼1 jkni j 6 C for each n. The problem which we will consider is to find sufficient conditions for strong e n g in the following; convergence of weighted sums of f X n n e e lim d 1 kni X i ; kni coðE X i Þ ¼ 0 a:s:; n!1
i¼1
i¼1
e i Þ denotes the convex hull of Eð X e i Þ. where coðE X To this end, we require the following assumption. For each > 0; there exists a partition 0 = a0 < a1 < < am = 1 of [0, 1] such that for all n, e n ; Lak X e n Þ < : max EhðLaþk1 X
16k6m
ð3:1Þ
e n g is identically distributed, then it satThe next theorem implies that if f X isfies the condition (3.1). e 1 k < 1, Then for each > 0; there exists a partition Theorem 3.1. (a) Let Ek X 0 = a0 < a1 < < am = 1 of [0, 1] such that e ; Lak X e Þ < : max EhðLaþ X k1
16k6m
e n g is identically distributed and Ek X e 1 k < 1, then it satisfies the condi(b) If f X tion (3.1). Proof. (a) Suppose that the assertion does not hold. Then we can find a 0 > 0 such that for any partition P : 0 ¼ a0 < a1 < < am ¼ 1 of [0, 1], there is a ak in P satisfying e ; Lak X e Þ P 0 : EhðLaþ X k1
We consider a sequence fPn g of partitions of [0, 1] such that (1) Pnþ1 is a refinement of Pn ; (2) kPn k ! 0, where kPn k denotes the longest length of largest subinterval of kPn k. For each n, let akn be the largest point of Pn such that e ; Lak X e Þ P 0 : EhðLaþ X k n 1
n
ð3:2Þ
Then fakn g is monotone decreasing since Pnþ1 is a refinement of Pn . By (3.2), we can choose a sequence {bn} with akn 1 < bn < akn such that for all n, e ; Lak X e Þ > 0 =2: EhðLbn X n
ð3:3Þ
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Since fakn g is monotone decreasing and kPn k ! 0, it follows that e ; Lak X e Þ ! 0 pointwise: hðLb X n
n
Then by the Lebesgue dominated convergence theorem, e ; Lak X e Þ ! 0; EhðLb X n
n
which is impossible by (3.3). (b) It follows from the fact that e n ; Lak X e n Þ ¼ EhðLaþ X e 1 ; Lak X e 1 Þ: EhðLaþk1 X k1
Now in order to obtain strong convergence of weighted sums of independent fuzzy random sets, we need the concept of independence for fuzzy random sets. By the way, there are several definitions of independence for fuzzy random sets. In this paper, we adopt the following notion of independence. e n g be a sequence of fuzzy random sets. Then f X e n g is Definition 3.2. Let f X e n Þg is independent, where rð X e Þ is the called independent if the r-fields frð X e : X ! ðF; d s Þ is measurable. smallest r-field of subsets of X such that X e j0 6 a 6 1gÞ because X e : X ! ðF; d s Þ is measure Þ ¼ rðfLa X Note that rð X e : X ! ðK; hÞ is measurable. able if and only if for each a 2 [0, 1], La X e n g be a sequence of independent fuzzy random sets Theorem 3.3. Let f X satisfying (3.1). Suppose that there exists a nonnegative random variable X with 1 EX 1þc < 1 for some c > 0 such that for each n, e n k P kÞ 6 P ðX P kÞ P ðk X
for all k > 0:
ð3:4Þ
If {kni} is a Toeplitz sequence satisfying max16i6njknij = O(nc), then n n e e lim d 1 kni X i ; kni coðE X i Þ ¼ 0 a:s: n!1
i¼1
i¼1
Proof. Since {kni} is a Toeplitz sequence, there exist a positive number C such that 1 X jkni j 6 C for all n: i¼1
Let > 0 be given. By assumption (3.1), we choose 0 = a0 < a1 < < am = 1 of [0, 1] such that for all n e n ; Lak X e n Þ < =C: max EhðLaþk1 X
16k6m
ð3:5Þ
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Then
S.Y. Joo et al. / Information Sciences 176 (2006) 1086–1099
n n e i ; kni La coðE X e iÞ sup h kni La X i¼1
ak1
i¼1
n n n n e e e e sup h kni La X i ; kni Lak X i þ h kni Lak X i ; kni Lak coðE X i Þ
6
i¼1
ak1
i¼1
n
n
i¼1
i¼1
i¼1
e i Þ; kni La coðE X e iÞ þh kni Lak coðE X 6
n X
n n e e e e jkni jhðLaþk1 X i ;Lak X i Þ þ h kni Lak X i ; kni Lak coðE X i Þ i¼1
i¼1
þ
i¼1
n X
i¼1
e i Þ;Lak coðE X e i ÞÞ ¼ ðIÞ þ ðIIÞ þ ðIIIÞ: jkni jhðLaþk1 coðE X
i¼1
For (I), we first note that for all n, e n ; Lak X e n Þ P kg 6 P f2k X e n k P kg 6 P ð2X P kÞ: P fhðLaþk1 X By RohatgiÕs result [18] for real-valued random variables, n X
e i ; Lak X e i Þ EhðLaþ X e i ; Lak X e i Þ ! 0 a:s: jkni j½hðLaþk1 X k1
i¼1
Thus by (3.5), n X e i ; Lak X e i Þ EhðLaþ X e i ; Lak X e i Þ ðIÞ ¼ jkni j½hðLaþk1 X k1 i¼1
þ
n X
e i ; Lak X e i Þ < a:s: for large n: jkni jEhðLaþk1 X
i¼1
e n kg is uniformly For (II), first we note that assumption (3.4) implies that fk X integrable. For, Z Z 1 e n k dP ¼ aP ðk X e n k > aÞ þ e n k > xÞ dx kX P ðk X fke X n k>ag a Z 1 Z 6 aP ðX > aÞ þ P ðX > xÞ dx ¼ X dP a
fX >ag
e n kg. Then it follows that for each a, implies the uniform integrability of fk X e fLa ; X n g is uniformly integrable. Recall that uniform integrability is equivalent to compactly uniform integrablity for a sequence of Rp-valued random sets. Thus we conclude from Theorem 3.3 of Taylor and Inoue [20] for random sets that ðIIÞ ! 0 a:s:
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Finally, for (III) by (3.5) and convexity inequality, we have n n X X e i ; co Lak X e iÞ 6 e i ; Lak X e i Þ < : jkni jEhðco Laþk1 X jkni jEhðLaþk1 X ðIIIÞ ¼ i¼1
i¼1
Therefore, n n e e d 1 kni X i ; kni coðE X i Þ i¼1 i¼1 n n e e ¼ max sup h kni La X i ; La kni coðE X i Þ < 2 a:s: for large n: 16k6m ak 1
i¼1
i¼1
Since is arbitrary, this completes the proof.
h
The following Corollary is an analogue of strong law of large numbers obtained by Mochanov [15]. e n g be a sequence of independent and identically distributed Corollary 3.4. Let f X 1 e 1 k1þc < 1 for some c > 0. If {kni} is a Toeplitz fuzzy random sets with Ek X sequence satisfying max16i6njknij = O(nc), then ! ! n X n e e lim d 1 kni X i ; kni coðE X 1 Þ ¼ 0 a:s: n!1
i¼1
i¼1
e n g be a sequence of independent fuzzy random sets satisfyCorollary 3.5. Let f X ing (3.1). If e n kr ¼ M < 1 sup Ek X
for some r > 1
ð3:6Þ
n
then for any Toeplitz sequence {kni} satisfying max16i6n jkni j ¼ Oðnc Þ for 1 c > r1 , n n e i ; kni coðE X e i Þ ¼ 0 a:s: lim d 1 kni X n!1
i¼1
i¼1
Proof. By Lemma 3 of Wei and Taylor [23], (3.6) implies that there exists a non1 negative random variable X with EX 1þc < 1 for 0 < 1c < r 1 such that (3.4) is satisfied for each n. Thus the result follows immediately from Theorem 3.3. h e n 2 Kg ¼ 1 Remark. If there exists a compact subset K of ðF; d s Þ such that P f X for all n, then (3.6) is also satisfied for all r > 1. Recall that we can define the concept of convexity on F as in the case of a vector space. That is, A F is said to be convex if ke u ð1 kÞev 2 A whenever
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e u , ev 2 A and 0 6 k 6 1. Using this concept of convexity on F, Kim [12] obtained the following; Theorem 3.6. Let us denote by CðFÞ the collection of all relatively compact subsets K of ðF; d s Þ for which the convex hull of K is also relatively compact. Then K 2 CðFÞ if and only if the following two conditions hold: (1) supfke uk : e u 2 Kg < 1. (2) For each > 0, there exists a finite partition 0 = a0 < a1 < < am = 1 of [0, 1], such that u; Lak e u Þ 6 : sup max hðLaþk1 ~ eu2K 16k6m By applying this result, we can obtain the following. e n g be a sequence of independent and convexly tight fuzzy Corollary 3.7. Let f X random sets, i.e., for each > 0, there exists a compact convex subset K of ðF; d s Þ such that e n 62 KÞ < for all n: P ðX e n k ¼ M < 1 for some r > 1, then for any Toeplitz sequence {kni} If supn Ek X 1 satisfying max16i6njknij = O(nc) for c > r1 , n n e i ; kni coðE X e i Þ ¼ 0 a:s: lim d 1 kni X r
n!1
i¼1
i¼1
e n g satisfies the assumption (3.1) in view of Proof. It suffices to prove that f X e n g, we can choose Corollary 3.5. Let > 0 be given. By convex tightness of f X a compact convex subset K of ðF; d s Þ such that e n 62 KÞ < ð=4Þr=ðr1Þ M 1=ðr1Þ P ðX
for all n:
Then e n kÞ 6 ðEk X e n kr Þ1=r P ð X e n 62 KÞðr1Þ=r EðI fe kX X n 62Kg
e n 62 2KÞðr1Þ=r < =4: 6 M 1=r P ð X
Now by Theorem 3.6, there exists a finite partition 0 = a0 < a1 < < am = 1 of [0, 1], such that sup max hðLaþk1 ~ u; Lak ~ uÞ 6 =2: eu2K 16k6m
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Thus, we have for all n, e n ; Lak X e n Þ 6 max EI e n ; Lak X e nÞ hðLaþk1 X max EhðLaþk1 X fe X n 2Kg 16k6m
16k6m
e n ; Lak X e nÞ þ max EI fe hðLaþk1 X X n 62Kg 16k6m e n k < : 6 =2 þ 2EI fe kX X n 62Kg h 1=n; if 1 6 i 6 n; then we can obtain a If we apply Theorem 3.3 to kni ¼ 0; if i > n; SLLN for independent fuzzy random sets. But in this case, we need the restrictive condition EjX j2 < 1. Similarly, we need the restrictive condition e n kr < 1 for some r > 2 in order to provide a SLLN by applying Corsupn Ek X ollaries 3.5 and 3.7. However, we can obtain much better SLLN by similar arguments as in the proof of Theorem 3.3. This yields the desired result.
e n g be a sequence of independent fuzzy random sets Theorem 3.8. Let f X satisfying (3.1). Suppose that the following two conditions are satisfied; e n kg is uniformly integrable. (1) fk X P1 1 e n kr < 1 for some 1 6 r 6 2. (2) n¼1 nr Ek X Then
n n 1 e i ; coðE X e i Þ ¼ 0 a:s: d1 X n i¼1 i¼1
lim
n!1
Proof. Let > 0 be given. By assumption (3.1), we choose 0 = a0 < a1 < < am = 1 of [0, 1] such that for all n, e n ; Lak X e n Þ < : max16k6m EhðLaþk1 X
ð3:7Þ
Now, 1 n
sup ak1
6
1 n
n n e i ; La coðE X e iÞ h La X i¼1
sup ak1
i¼1
n n n n e e e e h La X i ; Lak X i þ h Lak X i ; i¼1 Lak coðE X i Þ i¼1
i¼1
i¼1
n n e i Þ; La coðE X e iÞ þh Lak coðE X i¼1
i¼1
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n n n 1X 1 e e e e 6 hðLaþk1 X i ; Lak X i Þ þ h Lak X i ; Lak coðE X i Þ n i¼1 n i¼1 i¼1 n 1X e i Þ; Lak coðE X e i ÞÞ ¼ ðIÞ þ ðIIÞ þ ðIIIÞ: þ hðLaþk1 coðE X n i¼1 For (I), we first note that for all n, e n ; Lak X e n Þ 6 2r Ek X e nk : E½hðLaþk1 X r
r
By ChungÕs SLLN for real-valued random variables, n X 1 e i ; Lak X e i Þ EhðLaþ X e i ; Lak X e i Þ ! 0 a:s: ½hðLaþk1 X k1 n i¼1
Thus, by (3.7), ðIÞ ¼
n 1X e i ; Lak X e i Þ EhðLaþ X e i ; Lak X e i Þ ½hðLaþk1 X k1 n i¼1 n 1X e i ; Lak X e i Þ < a:s: for large n: þ EhðLaþk1 X n i¼1
e n g is compactly uniformly integrable, it follows from For (II), since fLak X SLLN for random sets (see [20]) that ðIIÞ ! 0 a:s: Finally, for (III) by (3.7), we have ðIIIÞ 6
n 1X e i ; La X e i Þ 6 : EhðLaþk1 X n i¼1
Therefore, n n 1 e i ; coðE X e i Þ ¼ 1 max d1 X n n 16k6m i¼1 i¼1
sup ak1
n n e i ; La coðE X e iÞ h La X i¼1
i¼1
6 2 a:s: for large n: Since is arbitrary, this completes the proof.
h
e n g be a sequence of independent fuzzy random sets satisfyCorollary 3.9. Let f X e n kg is tight ing (3.1). Suppose that a sequence of real-valued random variables fk X and e n k < 1 for some r > 1: sup Ek X r
n
S.Y. Joo et al. / Information Sciences 176 (2006) 1086–1099
Then
1097
n n 1 e e lim d 1 X i ; coðE X i Þ ¼ 0 a:s: n!1 n i¼1 i¼1
Proof. This follows from the fact that tightness and rth (r > 1) moments cone n kg imply uniform integrability of fk X e n kg. h dition of fk X e n g be a sequence of independent and convexly tight fuzzy Corollary 3.10. Let f X e n kr ¼ M < 1 for some r > 1, then random sets. If supn Ek X n n 1 e e lim d 1 X i ; coðE X i Þ ¼ 0 a:s: n!1 n i¼1 i¼1 e n g implies of tightness of fk X e n kg. Proof. It is trivial that convex tightness of f X e n g satisfies the assumption (3.1). h By the proof of Corollary 3.7, f X Finally, we will give an example in order to apply our results Example. Let e u 2 F be fixed. For each n, we let 1 e n ¼ n~ P ðX uÞ ¼ ; n
1 en ¼ ~ P ðX 0Þ ¼ 1 : n e e n ; Lb X e nÞ ¼ e Then it follows that Eð X n Þ ¼ e u and Ek X n k ¼ ke u k. Since EhðLa X e n g is not e n g satisfy (3.1). But f X u ; Lb e u Þ, it can be easily proved that f X hðLa e tight. For, if K is a compact subset of F, then there exists a n0 such that n0 e u 62 K. And so, Z e n k dP ¼ ke kX u k: sup n fe X n 62Kg Now let f Ye n g be independent fuzzy random sets such that for some r > 0, 1 P ð Ye n ¼ ne uÞ ¼ r ; n
1 P ð Ye n ¼ e 0Þ ¼ 1 r : n u k. Also, since u and Ek Ye n k ¼ n1r ke Then it can be shown that Eð Ye n Þ ¼ n1r e 1r EhðLa Ye n ; Lb Ye n Þ ¼ n hðLa e u ; Lb e u Þ, it satisfies the condition (3.1). Since u kr , by Corollary 3.5, Ek Ye n kr ¼ ke ! ! n X n k ni lim d 1 kni Ye i ; coðe u Þ ¼ 0 a:s: n!1 i¼1 ir1 i¼1 for any Toeplitz sequence {kni} satisfying max16i Also, since fk Ye n kg is tight, by Corollary 3.9,
6nkknik
1 = O(nc) for c > r1 .
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! ! n X n 1 1 lim d 1 Ye i ; coðe u Þ ¼ 0 a:s: n!1 n i¼1 ir1 i¼1
4. Concluding remarks In this study, we generalized the results of strong convergence for weighted sums of independent random sets presented by Taylor and Inoue [20]. It is still to be demonstrated whether similar results hold in the case assumption (3.1) has not been imposed. References [1] Z. Artstein, R.A. Vitale, A strong law of large numbers for random compact sets, Ann. Probab. 3 (1975) 879–882. [2] A. Colubi, J.S. Dominguez-Menchero, M. Lo´pez-Dia´z, M.A. Gil, A generalized strong law of large numbers, Probab. Theory Rel. Fields 114 (1999) 401–417. [3] A. Colubi, J.S. Dominguez-Menchero, M. Lo´pez-Dia´z, R. Ko¨rner, A method to derive strong laws of large numbers for random upper semicontinuous functions, Statist. Probab. Lett. 53 (2001) 269–275. [4] A. Colubi, J.S. Dominguez-Menchero, M. Lopez-Diaz, D. Ralescu, On the formalization of fuzzy random variables, Inform. Sci. 133 (2001) 3–6. [5] G. Debreu, Integration of correspondences, Proc. 5th Berkeley Symp. Math. Statist. Prob. 2 (1966) 351–372. [6] Y. Feng, An approach to generalize law of large numbers for fuzzy random variables, Fuzzy Sets Syst. 128 (2002) 237–245. [7] F. Hiai, H. Umegaki, Integral, conditional expectation and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149–182. [8] H. Inoue, A strong law of large numbers for fuzzy random sets, Fuzzy Sets Syst. 41 (1991) 285– 291. [9] S.Y. Joo, Y.K. Kim, Topological properties on the space of fuzzy sets, J. Math. Anal. Appl. 246 (2000) 576–590. [10] Y.K. Kim, Strong law of large numbers for random upper semicontinuous functions, Bull. Kor. Math. Soc. 39 (2002) 511–526. [11] Y.K. Kim, Measurability for fuzzy valued functions, Fuzzy Sets Syst. 129 (2002) 105–109. [12] Y.K. Kim, Compactness and convexity on the space of fuzzy sets II, Nonlinear Anal. Appl. 57 (2004) 639–653. [13] E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. Lond. Ser. A 407 (1986) 171–182. [14] R. Lowen, Convex fuzzy sets, Fuzzy Sets Syst. 3 (1980) 291–310. [15] I. Molchanov, On strong law of large numbers for random upper semicontinuous functions, J. Math. Anal. Appl. 235 (1999) 349–355. [16] W. Pruitt, Summability of independent random variables, J. Math. Mech. Appl. 15 (1966) 769– 776. [17] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409– 422. [18] V.K. Rohatgi, Convergence of weighted sums of independent random variables, Proc. Cambridge Philos. Soc. 69 (1971) 305–307.
S.Y. Joo et al. / Information Sciences 176 (2006) 1086–1099
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[19] R.L. Taylor, H. Inoue, A strong law of large numbers for random sets in Banach space, Bull. Inst. Math. Acad. Sin. 13 (1985) 403–409. [20] R.L. Taylor, H. Inoue, Convergence of weighted sums of random sets, Stoch. Anal. Appl. 3 (1985) 379–396. [21] R.L. Taylor, L. Seymour, Y. Chen, Strong laws of large numbers for fuzzy random sets, University of Georgia, Department of Statistics, Technical report 2000-7. [22] T. Uemura, A law of large numbers for random sets, Fuzzy Sets Syst. 59 (1993) 181–188. [23] D. Wei, R.L. Taylor, Convergence of weighted sums of tight random elements, J. Multivariate Anal. 8 (1978) 282–294.