Strong limit theorems for random sets and fuzzy ... - Semantic Scholar
Available online at www.sciencedirect.com
Information Sciences 178 (2008) 2648–2660 www.elsevier.com/locate/ins
Strong limit theorems for random sets and fuzzy random sets with slowly varying weights q Ke-ang Fu , Li-xin Zhang Department of Mathematics, Zhejiang University, Hangzhou 310027, China Received 4 November 2005; received in revised form 8 August 2007; accepted 9 January 2008
Abstract Theories of random sets and fuzzy random sets are useful concepts which are frequently applied in scientific areas including information science, probability and statistics. In this paper strong limit theorems are derived for random sets and fuzzy random sets with slowly varying weights in separable Banach spaces. Both independent and dependent cases are covered to provide a wide range of applications. Ó 2008 Elsevier Inc. All rights reserved. AMS classification: 60F15; 60B11 Keywords: Random set; Fuzzy random set; Slowly varying weights; Hausdorff distance; Dependence
1. Introduction In recent years, the theories of random sets and fuzzy random sets have been extensively studied and applied in the areas of information science, probability and statistics. The general idea of random sets has been in existence for some time. Robbins [26,27] appeared to be the first to provide the concept of random sets, and his early works investigated the relationships between random sets and geometric probabilities. Later, Kendall [13] and Matherson [21] provided a comprehensive mathematical theory of random sets which was greatly influenced by the geometric probability prospective. Their proposed framework exerted a strong influence on the limit theorems developed in recent decades. As applications in probability and statistics, statistical inference for random sets leads to the need of laws of large numbers to insure consistency in estimation problems. Hence, laws of large numbers play an important role in statistical inference, and for details one can refer to Artstein and Vitale [2], Puri and Ralescu [22,23], Taylor et al. [29–33], Uemura [35], and references therein. Among them, Artstein and Vitale [2] proved limit theorems concerning random sets in R and Rd ; and Puri and q
Project supported by National Natural Science Foundation of China (Nos. 10671176 & 10771192). Corresponding author. Tel.: +86 13777398392. E-mail addresses: [email protected] (K.-a. Fu), [email protected] (L.-x. Zhang).
0020-0255/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2008.01.005
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
2649
Ralescu [22] were the first to obtain the strong laws of large numbers (SLLN’s) for independent identically distributed (i.i.d.) Banach space-valued compact convex random sets. Among others, SLLN’s were obtained under more relaxed conditions, and a detailed survey of these results is available in Taylor and Inoue [31]. The theory of fuzzy sets was introduced by Zadeh [37] (for an outline recently, one can refer to [38]), and the concept of fuzzy random variables was promoted by Kwakernaak [17] where useful basic properties were developed. Puri and Rasescu [24] used the concept of fuzzy random variables to generalize the results of random sets to fuzzy random sets. With respect to laws of large numbers, Kruse [16] proved an SLLN for i.i.d. fuzzy random variables, and then gave a consistent estimator for the expectation of a fuzzy random variable as an application of the SLLN. Klement et al. [14] considered fuzzy versions of random sets in Euclidean spaces and obtained an i.i.d SLLN. Inoue [11] obtained SLLN’s for independent, tight fuzzy random sets and i.i.d. fuzzy random sets in a separable Banach space. Recently, SLLN’s have been studied under various conditions, and one can refer to the following papers [6,9,12,15,18,19,30,31,34], and references therein. Also for more detailed results about limit theorems of random sets and fuzzy random sets, one can refer to Li et al. [20]. In this article, we establish some strong limit theorems (i.e. strong laws of large numbers) for i.i.d. random sets and fuzzy random sets with slowly varying weights, and we also obtain the sufficient and necessary conditions. The layout of this paper is as follows. In Section 2, we list the basic definitions and properties which will be used to obtain strong limit theorems. In Section 3, we state the main results of this paper and their detailed proofs are given in Section 4. In the last Section we show that our results are also true for a broad class of dependent random sets and fuzzy random sets. In the sequel, let C; C 0 and M denote a positive constant whose value can differ in different places. 2. Definitions and preliminaries Throughout this paper, let S be a real separable Banach space with the norm k k and the dual space S . For each A S; clA and coA denote the norm-closure and the closed convex hull of A, respectively. Let CðSÞ (resp. C c ðSÞ) denote the collections of all non-empty compact (resp. non-empty closed compact) subsets of S. Define the Minkowski’s addition and scalar multiplication, respectively, in CðSÞ (or C c ðSÞ) by A þ B ¼ fa þ bja 2 A; b 2 Bg; kA ¼ fkaja 2 Ag; where A; B 2 CðSÞ (or C c ðSÞ) and k is a real number. Neither CðSÞ nor C c ðSÞ are linear spaces even when S ¼ R, one-dimensional Euclidean space. And for A; B 2 CðSÞ, the Hausdorff distance d H ðA; BÞ of A and B and the norm kAk of A are defined by d H ðA; BÞ ¼ maxfsup inf ka bk; sup inf ka bkg; a2A b2B
b2B a2A
kAk ¼ d H ðA; f0gÞ ¼ sup kak: a2A
Let ðX; F; PÞ denote a probability measure space. A random (compact) set is a Borel measurable function F : X ! CðSÞ; that is, F 1 ðBÞ ¼ fx 2 X; F ðxÞ \ B 6¼ ;g 2 F for each B 2 CðSÞ (c.f. [10,20]). For a random set F in CðSÞ; there exists a corresponding set coF in C c ðSÞ; which can be used in defining an expected value. A measurable function f : X ! S is called a measurable selection of F if f ðxÞ 2 F ðxÞ for every x 2 X. Denote by S F ¼ ff 2 L1 ðX; SÞ; f ðxÞ 2 F ðxÞ; a:e:g;
R where L1 ðX; SÞ denotes the space of measurable functions f : X ! S such that n kf ðxÞkdP < 1. S F 6¼ ; if and only if the random variable kF ðxÞk is integrable. The random set F is called integrably bounded if the realvalued random variable kF ðxÞk is integrable (c.f. [10,20]). Hiai and Umegaki [10] showed that a random set F is integrably bounded if and only if S F is bounded in L1 ðX; SÞ. Thus an integrably bounded random set may take unbounded sets.
2650
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
For each random set F, the expectation of F, denoted by EðF Þ; is defined by Z Z EðF Þ ¼ F dP ¼ f dP; f 2 S F ; X
ð1Þ
X
R R R where X f dP is the usual Bochner integral in L1 ðX; SÞ. Define A F dP ¼ A f dP; f 2 S F for A 2 F. This definition was introduced by Aumann in 1965 as a natural generalization of the integral of real-valued random variables Rin [3]. For the random set F, if EkcoF k < 1; then a Bochner integral can be defined as EðcoF Þ ¼ X coF dP and EðcoF Þ 2 C c ðSÞ [7]. For A, B, C and D 2 CðSÞ; k 2 R and X ; Y are random sets, immediate properties of the Hausdorff distance lead to the following: d H ðkA; kBÞ 6 jkjd H ðA; BÞ; d H ðA; BÞ 6 d H ðA; CÞ þ d H ðC; BÞ; d H ðA þ C; B þ DÞ 6 d H ðA; BÞ þ d H ðC; DÞ; d H ðcoA; coBÞ 6 d H ðA; BÞ; d H ðEcoX ; EcoY Þ 6 Ed H ðX ; Y Þ: A finite set of random sets fX 1 ; . . . ; X n g in CðSÞ is said to be independent if PðX 1 2 B1 ; . . . ; X n 2 Bn Þ ¼ PðX 1 2 B1 Þ PðX n 2 Bn Þ for every B1 ; . . . ; Bn 2 CðSÞ. A family of random set in CðSÞ is said to be independent if every finite subset is independent. The random sets X and Y are said to be identical distributed if PðX 2 BÞ ¼ PðY 2 BÞ for every B 2 CðSÞ. A family of random sets is identically distributed if every pair is identically distributed. From the definitions above, we know that most stochastic properties of random sets are extended from real-valued random variables directly. Now we begin to state some properties about fuzzy sets. A fuzzy set in S is a function u : S ! ½0; 1. Let F ðSÞ denote the family of the fuzzy subset u satisfying the following conditions: (a) u is upper semicontinuous, that is, the a-level set of u, i.e. ua ¼ fx 2 S; uðxÞ P ag is a closed subset of S for each a 2 ð0; 1, (b) fx 2 S : uðxÞ > 0g has compact closure, (c) fx 2 S : uðxÞ ¼ 1g 6¼ ;. For u 2 F ðSÞ; the support of u is defined as supp u ¼ fx 2 S : uðxÞ > 0g, and the assumption of compact closure implies that support of u is norm bounded. A linear structure in F ðSÞ is defined by the following operations: ðu þ vÞðxÞ ¼ sup min½uðyÞ; vðzÞ; yþz¼x 1 uðk xÞ; if k 6¼ 0; ðkuÞðxÞ I 0 ðxÞ; if k ¼ 0; where u; v 2 F ðSÞ; k 2 R; and I 0 ðÞ is an indicator function. The linear structure aforementioned implies the following properties: ðu þ vÞa ¼ ua þ va ; ðkuÞa ¼ kua : And for fuzzy sets, we adopt the most common metric d r , viewed as a generalization of the Hausdorff metric from CðSÞ to F ðSÞ; where Z 1 1=r d r ðu; vÞ ¼ d rH ðua ; va Þda ; ð2Þ 0
where u; v 2 F ðSÞ and 1 6 r 6 2. Also we define a topology T using the countable family of semimetrics d~a ðu; vÞ ¼ d H ðua ; va Þ;
a2 W;
where W denotes the dyadic rational numbers, i.e. W ¼ f2im : 1 6 i 6 2m ; m P 1g.
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
2651
Definition 2.1. The topology T is defined by the smallest which makes each projection La , a 2 W ; P 1topology d~ai ðu;vÞ continuous. The topology is meterizable by d T ¼ 1 for all u; v 2 F ðSÞ; where a1 ; a2 ; . . . is a i i¼1 2 1þd~a ðu;vÞ i countable listing of W. Since every semimetric d~a is separable for F ðSÞ; it follows that T is separable. Taylor et al. [32] showed that T and d r generate the same Borel subsets, and hence have the same fuzzy random sets defined as below. The concept of a fuzzy random set as a generation for a random set was extensively studied by Puri and Ralescu [24]. A fuzzy random set is a function X : X ! F ðSÞ such that for each a 2 ð0; 1, X a ðxÞ ¼ fx 2 S; X ðxÞðxÞ P ag is a random set in S (c.f. [20]). A random fuzzy set X is said to be integrably bounded if the real-valued random variable ksupp uk is integrable. The expectation of a fuzzy random set X, denoted by E½X ; is an element in F ðSÞ such that for each a 2 ð0; 1; Z ðE½X Þa ¼ cl X a dP ¼ clfEðf Þ; f 2 S X a g; X
where the closure is taken in S and S X a ¼ ff 2 L1 ðX; SÞ; f ðxÞ 2 X a ðxÞ a:e:g. By virtue of the existence theorem (c.f. [20]), we have an equivalent definition as follows: E½X ðxÞ ¼ supfa 2 ð0; 1; x 2 E½X a g: Furthermore, ðE½coX Þa ¼ E½ðcoX Þa for any a 2 ð0; 1. A sequence fuzzy random sets fX n ; n P 1g is said to be independent if for any a 2 ð0; 1; the sequence of random sets fX n a; n P 1g is independent, and the identical distribution can be defined similarly. Now we introduce something about slowly varying functions. Definition 2.2. A monotone (increasing or decreasing) function L : ½a; 1Þ ! ð0; 1Þ, where a P 0, is said to be slowly varying if lim
x!1
LðtxÞ ¼1 LðxÞ
8t > 0:
ð3Þ
It is well known [28] that if L is slowly varying, then limx!1 Lðx þ 1Þ=LðxÞ ¼ 1 and LðxÞ ¼ Oðxi Þ and 1=LðxÞ ¼ Oðxi Þ as x ! 1; for any i > 0:
ð4Þ
Also P if L is a slowly varying function defined on ½1; 1Þ, then it follows from (4) and Theorem 1(b) of Feller [8] n that j¼1 LðjÞ nLðnÞ % 1. 3. Main results In this section, we list the strong laws of large numbers for random sets and fuzzy random sets, respectively. The first one is for independent and identical distributed (i.i.d.) random sets, and we establish the sufficient and necessary conditions. Theorem 3.1. Let fX n ; n P 1g be a sequence of i.i.d. random sets in CðSÞ, (i) let L be a slowly varying function defined on ½1; 1Þ, and let m 2 C c ðSÞ, then Pn j¼1 LðjÞcoX j ! m a:s: ð5Þ Rn1 :¼ nLðnÞ if and only if EðkcoX 1 kÞ < 1
and
EcoX 1 ¼ m;
ð6Þ
(ii) Moreover, Pn Rn2 :¼
j¼1 LðjÞX j
nLðnÞ
!m
a:s:
ð7Þ
if and only if (6) holds, where the convergence is in the Hausdorff metric d H of CðSÞ and co denotes the continuous mapping of CðSÞ into C c ðSÞ.
2652
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
Remark 3.1. If the identical distribution hypothesis is relaxed to that the random sets are (uniformly) tight and stochastically dominated by a random variable X, then the conclusions above remain true. Remark 3.2. Taking LðjÞ ¼ C 0 (a fixed constant) yields the following Corollary which has been proved with various approaches. Corollary 3.1. Let fX n ; n P 1g be a sequence of i.i.d. random sets in CðSÞ, such that EðkcoX 1 kÞ < 1 and EðcoX 1 Þ ¼ m. Then in Hausdorff sense n 1X X j ! m a:s: ð8Þ n j¼1 The following result is for i.i.d. fuzzy random sets, but we only give the sufficient condition. Theorem 3.2. Let fX n ; n P 1g be a sequence of i.i.d. fuzzy random sets in F ðSÞ such that EkX 1 kr < 1, and let Rn2 be defined in form of (7). Then we have d r ðRn2 ; EX 1 Þ ! 0 a:s:;
ð9Þ
where r is defined by the metric d r in (2). r
Conjecture 3.1. We believe that EkX 1 k < 1 is still the necessary condition for fuzzy random sets. To get such an improvement of the results, we think a different approach is necessary. 4. Proofs In this section, we will give the detailed proof for the two theorems above. And to prove the theorems, we need to introduce the following lemmas. Lemma 4.1 [1]. Let fX n ; n P 1g be real independent random variables and let X be a real random variable with EjX jp < 1 for some 1 6 p < 2. Suppose that fX n ; n P 1g is stochastically dominated by X in the sense that there exits a constant D < 1 such that PðjX n j > tÞ 6 DPðjDX j > tÞ;
t P 0; n P 1: 1
Let fan ; n P 1g and fbn ; n P 1g be constants satisfying 0 < bn " 1, an =bn ¼ Oðnp Þ and we have the SLLN: n X aj ðX j EX j Þ ! 0 a:s: b1 n
Pn
j¼1 jaj j
¼ Oðbn Þ. Then
j¼1
Lemma 4.2 [1]. Let X 0 and X be real random variables such that X 0 is stochastically dominated by X in the sense that PðjX 0 j > tÞ 6 DPðjDX j > tÞ holds for all t > 0. Then Z 1 EjX 0 jIðjX 0 j > xÞ ¼ PðjX 0 j > tÞdt þ xPðjX 0 j > xÞ; x P 0 x
and EjX 0 jIðjX 0 j > xÞ 6 D2 EjX jIðjDX j > xÞ;
x P 0:
The following two lemmas are only slight modifications of the proofs in [1]. They are stated here for the convenience of the readers. Lemma 4.3. Let fV n ; n P 1g be independent, mean zero random elements in a separable Banach space admitting a Schauder basis fbi g with the coordinate functional ffi g, and let fan ; n P 1g and fbn ; n P 1g be defined as those
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
2653
P P1 in Lemma 4.1. Denote U m ðV Þ ¼ m i¼1 fi ðV Þbi and Qm ðV Þ ¼ i¼mþ1 fi ðV Þbi ; m P 1. Suppose that there exist real random variables fX i ; i P 1g and fY m ; m P 1g and a constant D < 1 such that for some 1 6 p < 2, Pðjfi ðV n Þj > tÞ 6 DPðjDX i j > tÞ;
t P 0; n P 1; i P 1;
PðjkQm ðV n Þ EkQm ðV n Þkj > tÞ 6 DPðjDY m j > tÞ; p
t P 0; n P 1; m P 1;
p
sup EjX i j < 1;
sup EjY m j < 1
iP1
mP1
and lim sup EkQm ðV n Þk ¼ 0:
ð10Þ
m!1 nP1
Then we have the SLLN n X aj V j ! 0 a:s b1 n j¼1
. Proof. It follows immediately from Lemma 4.1 that n X b1 aj fi ðV j Þ ! 0 a:s: for each i P 1 n
ð11Þ
j¼1
and T mn :¼ b1 n
n X
jaj jðkQm ðV j Þk EkQm ðV j ÞkÞ ! 0
a:s: for each m P 1:
ð12Þ
j¼1
Also it follows that ! ! X X n m n m n X X 1 X 1 1 aj V j ¼ fi bn aj V j b i 6 aj fi ðV j Þ kbi k: U m b n bn j¼1 i¼1 j¼1 i¼1 j¼1 Pn Thus, by (11)–(13) and j¼1 jaj j ¼ Oðbn Þ, we have ! ! X n n n X X b1 aj V j 6 U m b1 aj V j þ Qm b1 aj V j n n n j¼1 j¼1 j¼1 ! X n n X aj V j þ b1 aj Qm ðV j Þ 6 U m b1 n n j¼1 j¼1 ! n X 6 U m b1 aj V j þ T mn þ C sup EkQm ðV j Þk ! 0 a:s: n jP1 j¼1
ð13Þ
as m ! 1 after letting n ! 1. h Lemma 4.4. Let fV n ; n P 1g be a (uniformly) tight sequence of independent, mean zero random elements in a Banach space E, and let fan ; n P 1g and fbn ; n P 1g be defined as those in Lemma 4.1. Suppose that p fV nP ; n P 1g is stochastically dominated by a random element V and EkV k < 1 for some 1 6 p < 2. Then n 1 bn j¼1 aj V j ! 0 a.s. Proof. Since the isometrically embedding of E into a Banach space which has a monotone Schauder basis preserves the independence and uniformly tightness, we might as well assume that E has a monotone basis where kQm ðyÞk 6 kyk and kfm ðyÞk 6 kyk for each y 2 E and m P 1 and fkQm ðyÞk; m P 1g is a monotone decreasing sequence for each y 2 E. Now, for 8e > 0, choose u > 0 such that D2 EkV kIðkV k > uÞ 6 e=3:
2654
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
Thus, from Lemma 4.2, it follows EkV n kIðkV n k > uÞ 6 e=3 for all n P 1. By (uniformly) tightness, we can choose a compact subset K such that PðV n 62 KÞ 6 e=ð3uÞ. Thus it yields EkV n kIðkV n k 6 uÞIðV n 62 KÞ 6 e=3 for all n P 1. Since kQm ðzÞk # 0 for all z 2 K, there exists an integer m0 such that supz2K kQm ðzÞk 6 e=3 for all m P m0 . Hence, for all m P m0 and n P 1, EkQm ðV n Þk 6 EkQm ðV n ÞIðkV n k 6 uÞIðV n 2 KÞk þ EkV n kIðkV n k 6 uÞIðV n 62 KÞ þ EkV n kIðkV n k > uÞ 6 e: Since e is arbitrary, we get that limm!1 supnP1 EkQm ðV n Þk ¼ 0. Now, Pn let X i ¼ kV i k, and Y m ¼ kV m k þ DEkV m k for all i P 1; m P 1. Applying Lemma 4.3, we have that b1 n k j¼1 aj V j k ! 0 a:s. h Remark 4.1. From the proof above, we can obtain the following conclusion which will be used later: for 8e > 0, we can find a compact subset K such that EkV n IðV n 62 KÞk 6 e:
ð14Þ
If fV n ; n P 1g is a sequence of identically distributed random elements, the condition of stochastic domination is automatically satisfied with V ¼ V 1 and D ¼ 1. Furthermore, identical distribution property ensures that fV n ; n P 1g is automatically (uniformly) tight [29]. Hence, we have the following proposition. Proposition 4.5. Let fV n ; n P 1g be a sequence of i.i.d. random elements in a separable Banach space E, and let p fan ; nPP 1g and fbn ; n P 1g be defined as that in Lemma 4.1. If EkV n k < 1 for some 1 6 p < 2, then n 1 bn k j¼1 aj ðV j EV j Þk ! 0 a:s. Ra˚dstro¨m [25] showed that the collection of compact convex subsets of a Banach space can be embedded as a convex cone in a normed linear space. That is, the metric space C c ðSÞ is embedded in a separable Banach space N with an isometry g : C c ðSÞ ! N. Lemma 4.6 [25]. Let X : X ! C c ðSÞ be a random set such that EkX k < 1. If g : C c ðSÞ ! N is the isometry given by the Ra˚dstro¨P m embedding P theorem, then Eðg X Þ ¼ gðEX Þ. In particular, if xi 2 C c ðSÞ, ai P 0, i ¼ 1; 2; . . . ; n, then gð ni¼1 ai xi Þ ¼ ni¼1 ai gðxi Þ. Lemma 4.7 [30]. Let A 2 CðSÞ and fank g be an array of non-negative constants such that n X ank 6 1; ðiiÞ max ank ! 0 as n ! 1: ðiÞ k¼1
Then d H ð
Pn
k¼1 ank ck A;
16k6n
Pn
k¼1 ank ck coAÞ
! 0 for any sequence fcn g consisting of 1’s and 0’s.
Now, we state the proof of Theorem 3.1. Proof of Theorem 3.1. (i) Sufficiency. Assume that (6) holds and let g be defined as that in Lemma 4.6. Let an ¼ LðnÞ and P bn ¼ nj¼1 LðjÞ; n P 1. Then bn " 1. Also we know that bn nLðnÞ and abnn ¼ Oð1nÞ from preliminaries. By the distance preserving property of g and Lemma 4.6, ! ! ! n n n n X X X X aj coX j ; b1 aj EcoX j ¼ g b1 aj coX j g b1 aj EcoX j d H b1 n n n n j¼1 j¼1 j¼1 j¼1 n X
ð15Þ aj gðcoX j Þ EgðcoX j Þ : ¼ b1 n j¼1 Note that g coX ¼ gðcoX Þ is a random element in Banach space N, and EkgðcoX Þk ¼ EkcoX k < 1. Thus it follows from Proposition 4.5 that
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
n 1 X
aj gðcoX j Þ EgðcoX j Þ ! 0 bn j¼1
2655
a:s:
Hence, by applying (15), we have ! ! ! n n n n X X X X 1 1 1 1 d H bn LðjÞcoX j ; m ¼ d H bn LðjÞcoX j ; EcoX 1 ¼ d H bn aj coX j ; bn aj EcoX j j¼1
j¼1
!0
j¼1
j¼1
a:s:
Pn LðjÞcoX j ! m a.s. with respect to d H . Since bn nLðnÞ, we can easily get that Rn1 ¼ j¼1nLðnÞ Necessity. Assume that (5) holds. Then we have Pn Pn Pn
j¼1 LðjÞgðcoX j Þ j¼1 LðjÞcoX j j¼1 LðjÞcoX j gðmÞ ¼ g ;m ! 0 gðmÞ ¼ d H nLðnÞ nLðnÞ nLðnÞ Thus it follows that Pn Pn1 gðcoX n Þ ðn 1ÞLðn 1Þ j¼1 LðjÞgðcoX j Þ j¼1 LðjÞgðcoX j Þ ¼ ! gðmÞ 1 gðmÞ ¼ 0 n nLðnÞ nLðnÞ ðn 1ÞLðn 1Þ
a:s:
a:s:
in the sense that kcoX n k=n ¼ kgðcoX n Þk=n ! 0 a:s. Hence, we have PðkcoX n k=n > 1; i:o:ðnÞÞ ¼ 0. And coupled with the i.i.d. hypothesis and the Borel–Cantelli lemma, it leads to 1 1 X X PðkcoX 1 k > nÞ ¼ PðkcoX n k > nÞ < 1: n¼1
n¼1
Then it follows EkcoX n k ¼ EkcoX 1 k < 1; i.e., EcoX 1 exists. By the sufficiency partition of the theorem, we have Rn1 ! EcoX 1 a:s:; and hence together with (4) it implies EcoX 1 ¼ m. Thus the proof of part (i) is completed. (ii) Sufficiency. Since fX n ; n P 1g is i.i.d., as explained in Remark 4, for any e > 0, one can choose a compact subset K of CðSÞ such that EkX n IðX n 62 KÞk ¼ EkX 1 IðX 1 62 KÞk 6 e for all n, and since K is compact, there exist k 1 ; . . . ; k m 2 K such that m m [ [ Bðk j ; eÞ: K fz : kz k j k < eg
j¼1
j¼1
Z 0n IðX n
2 KÞ be a random set taking finitely many values, where Z 0n ¼ k 1 I½X n 2 Bðk 1 ; eÞþ Let Pm Z n ¼ Si1 c i¼2 k i IfðX Pnn 2 Bðk i ; eÞÞ \ l¼1 ðX n 2 Bðk l ; eÞ Þg. Thus fZ n ; n P 1g is also a sequence of i.i.d. random sets. Let bn ¼ j¼1 LðjÞ; and then we have Pn Pn
Pn
j¼1 LðjÞX j j¼1 LðjÞX j j¼1 LðjÞEcoX 1 ; m ¼ dH ; dH bn bn bn Pn Pn
j¼1 LðjÞX j j¼1 LðjÞX j IðX j 2 KÞ 6 dH ; bn bn P Pn Pn n
Pn
j¼1 LðjÞX j IðX j 2 KÞ j¼1 LðjÞZ j j¼1 LðjÞZ j j¼1 LðjÞEcoZ 1 þ dH ; ; þ dH bn bn bn bn þ d H ðEcoZ 1 ; EcoX 1 IðX 1 2 KÞÞ þ d H ðEcoX 1 IðX 1 2 KÞ; EcoX 1 Þ ¼: ðIÞ þ ðIIÞ þ ðIIIÞ þ ðIVÞ þ ðVÞ: Now we begin to deal with (I)–(V), respectively. For (I), Pn j¼1 LðjÞ kX j IðX j 62 KÞk: ðIÞ 6 bn
2656
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
Since fkX n IðX n 62 KÞk EkX n IðX n 62 KÞkg is a sequence of i.i.d. random variables with zero means, by Lemma 4.1 we can get that
Pn j¼1 LðjÞ kX j IðX j 62 KÞk EkX j IðX j 62 KÞk ! 0 a:s: bn Thus it follows that n X ðIÞ 6 b1 LðjÞkX j IðX j 62 KÞk ! EkX 1 IðX 1 62 KÞk 6 e a:s: n j¼1
As to (II), by the construction of Z n ; it implies n X LðjÞd H ðX j IðX j 2 KÞ; Z j Þ 6 e: ðIIÞ 6 b1 n j¼1
With regard to (III), we have Pn Pn
Pn
j¼1 LðjÞZ j j¼1 LðjÞcoZ j j¼1 LðjÞcoZ j ðIIIÞ 6 d H ; ; EcoZ 1 ¼: ðIII1 Þ þ ðIII2 Þ: þ dH bn bn bn Here we use Lemma 4.7 for cj ¼ IðZ j ¼ k i Þ and A ¼ k i . Since the number m is finite, we then obtain Pm Pm Pn Pn
j¼1 LðjÞ i¼1 k i IðZ j ¼ k i Þ j¼1 LðjÞ i¼1 cok i IðZ j ¼ k i Þ ðIII1 Þ ¼ d H ; bn bn Pn Pn
m X j¼1 LðjÞk i IðZ j ¼ k i Þ j¼1 LðjÞcok i IðZ j ¼ k i Þ 6 dH ; ! 0 a:s: bn bn i¼1 And ðIII2 Þ follows directly from the first part of Theorem 3.1. For (IV), from the construction of Z 1 and preliminaries, it follows that d H ðEcoZ 1 ; EcoX 1 IðX 1 2 KÞÞ 6 Ed H ðZ 1 ; X 1 IðX 1 2 KÞÞ 6 e: And about (V), we have d H ðEcoX 1 IðX 1 2 KÞ; EcoX 1 Þ 6 Ed H ðX 1 IðX 1 2 KÞ; EX 1 Þ ¼ EkX 1 IðX 1 62 KÞk 6 e: Thus by the arbitrariness of e, we have that ! n X 1 d H bn LðjÞX j ; m ! 0 a:s: j¼1 Pn LðjÞX j and hence, it follows that Rn2 ¼ j¼1 ! m a:s. with respect to d H due to bn nLðnÞ. nLðnÞ Necessity. Assume that (7) holds. Then we have Pn j¼1 LðjÞX j nLðnÞ ! kmk a:s: Notice that kAk 6 kA þ Bk þ kBk for sets A and B. It follows that # Pn1 "Pn j¼1 LðjÞX j ðn 1ÞLðn 1Þ LðjÞX j kX n k j¼1 þ lim sup 6 lim ¼ 2kmk a:s: n!1 n nLðnÞ nLðnÞ ðn 1ÞLðn 1Þ n!1 Since Pðkmk 6 MÞ > 0 for some constant M, we conclude that P kXnn k > 3M; i:o:ðnÞ < 1. By the i.i.d. hypothesis and the Borel–Cantelli lemma, we have 1 1 X X PðkX 1 k > 3MnÞ ¼ PðkX n k > 3MnÞ < 1: n¼1
n¼1
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
2657
Thus it follows that EkcoX 1 k 6 EkX 1 k < 1, i.e., EcoX 1 exists. By the sufficiency partition of the theorem, it leads to Rn2 ! EcoX 1 a:s. Hence, we have EcoX 1 ¼ m via (7). The proof is now terminated. Before proving Theorem 3.2, we need two more lemmas related to fuzzy random sets. Lemma 4.8 [32]. If supn ksupp un k < 1 and un ! u in T, then we have that for 1 6 r 6 2 d r ðun ; uÞ ! 0
as n ! 1:
Lemma 4.9 [32]. Let X be a fuzzy random set in F ðSÞ such that EkX krr < 1. Then we have Z
1
lim
M!1
EkX aIðkX ak > MÞkr da ¼ 0;
r P 1:
0
Now we turn to the Proof of Theorem 3.2. For every e > 0, choose an M > 0 such that Z 1 r EkX 1 aIðkX 1 ak > MÞk da < er 0
and for P every a 2 ½0; P1, set Ak a ¼ X k aIðkX k ak 6 MÞ and Bk a ¼ X k aIðkX k ak > MÞ. Note that X k a ¼ Ak a þ Bk a and ð nk¼1 X k Þa ¼ nk¼1 ðX k aÞ. Thus it follows that X k a0 s, Ak a0 s and Bk a0 s are all i.i.d. random sets due to the fact that X 0k s are i.i.d. And hence we have Z 1 1=r Z 1 Pn
1=r j¼1 LðjÞX j a r ; EcoðX 1 aÞ da d r ðRn2 ; EX 1 Þ ¼ d H ðRn2 a; EðX 1 aÞÞda ¼ d rH nLðnÞ 0 0 Z 1 Pn
1=r j¼1 LðjÞðAj a þ Bj aÞ ; EcoðA1 a þ B1 aÞ da ¼ d rH nLðnÞ 0 Z 1 Pn
1=r Z 1 Pn
1=r j¼1 LðjÞAj a j¼1 LðjÞBj a r r 0 0 ; EcoðA1 aÞ da ; EcoðB1 aÞ da 6C dH þC dH nLðnÞ nLðnÞ 0 0 ¼: C 0 ððVIÞ þ ðVIIÞÞ: For (VI), since kAk ak 6 M, Theorem 3.1 provides that for every a 2 W ; Pn
r j¼1 LðjÞAj a ; EcoðA1 aÞ dH ! 0 a:s: as n ! 1: nLðnÞ Since the dyadic rationales W are countable and fAk ag are bounded by M, it follows from Lemma 4.8 that
Pn
Z 1 Pn Z 1 j¼1 LðjÞAj a j¼1 LðjÞAj a r r ; EcoðA1 aÞ da ¼ ; EcoðA1 aÞ da ¼ 0 a:s: lim dH lim d H ð16Þ n!1 0 nLðnÞ nLðnÞ 0 n!1 For (VII), we have Z
1
d rH
0
R1
Pn
j¼1 LðjÞBj a
nLðnÞ
1=r
1=r (Z ; EcoðB1 aÞ da 6
Pn )1=r Z 1 1=r j¼1 LðjÞBj ar r da þ EkB1 ak da nLðnÞ 0 0 Z 1
1=r Xn 1 r 6 ðnLðnÞÞ LðjÞ kBj ak da þ e: j¼1
1
0
Since kBj akr da ;j P 1 is a sequence of i.i.d. random variables with Eð 0 R1 r r EkB1 ak da < e , by Lemma 4.1, we have 0
R1 0
kB1 akr daÞ ¼
2658
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
ðnLðnÞÞ
1
n X
Z LðjÞ
1 r
1=r
kBj ak da
!E
Z
0
j¼1
1 r
kB1 ak da
1=r
Z
1 r
EkB1 ak da
6
0
1=r <e
a:s:
0
Consequently, we obtain the desired relationship (9) by letting e ! 0. 5. Dependent random sets and fuzzy random sets In this section we show that our main results can be generalized to the case of dependent random sets and fuzzy random sets. When the sequence is not independent, we assume the following assumption instead. Assumption A. Let fX n ; n P 1g be a sequence of random variables (random elements, random sets or fuzzy random sets). Suppose for any sequence fY i ; i P 1g of real random variables with Y i being rðX i Þ measurable and EjY i j2 < 1, there exists a constant C > 0 such that 2 j X m X E max ðY i EY i Þ 6 C VarðY i Þ; m P n P 1: n6j6m i¼n i¼n When X 1 ; X 2 ; . . . are independent, Assumption A is obviously satisfied with C ¼ 2. This Assumption is also satisfied for many dependent random variables. For example, if fX n ; n P 1g is a u-mixing sequence wimixing P coefficient satisfying i uðiÞ < 1, Assumption A is satisfied. Mixing sequences are very popular in modeling dependent stochastic phenomena. Recently, Bradley [4], Byrc and Smolenjki [5], Utev and Peligrad [36] etc. studied a particular type of weakly mixing random variables defined as follows. For a random sequence fX n ; n P 1g, denote FT ¼ rðX i ; i 2 T Þ where T is a family of positive integers. Define qn ¼ sup
sup
jCorrðf ; gÞj;
f 2L2 ðFS Þ;g2L2 ðFT Þ
where the first sup is taken over all pairs of non-empty finite sets S, T of positive integers such that distðS; T Þ P n. Obviously, 0 6 qn 6 1. Many limit theorems are obtained for real random variables under the weak dependence condition that limn!1 qn < 1 (c.f., [4,5,36]). This weak dependence condition means that the random variables are not highly correlated when their distance is large enough. According to Theorem 2.1 of Utev and Peligrad [36], a random sequence fX n ; n P 1g satisfies Assumption A if limn!1 qn < 1. Theorem 5.1. Theorems 3.1 and 3.2 remain true with the conditions unchanged except that the condition X 1 ; X 2 ; are independent is replaced by Assumption A. The proof is similar to the one of Theorems 3.1 and 3.2 if Lemma 4.1 and the converse part of the Borel– Cantelli lemma are replaced by Lemmas 5.1 and 5.2 below, respectively. Lemma 5.1. Suppose Assumption A is satisfied and Y, Y 1 ; Y 2 ; . . . are random variables such that Y i is rðX i Þ measurable and EjY jp < 1 for some 1 6 p < 2. Assume that fY n ; n P 1g is stochastically dominated by1 Y. Let fan ; n P 1g and fbn ; n P 1g be sequences of constants satisfying 0 < bn " 1, an =bn ¼ Oðnp Þ and P n j¼1 jaj j ¼ Oðbn Þ. Then we have the SLLN: n X aj ðY j EY j Þ ! 0 a:s: b1 n j¼1
Lemma 5.2. Suppose Assumption A is satisfied. Let fAi ; i P 1g be a sequence of events such that Ai 2 rðX i Þ. Then PðAn ; i:o:ðnÞÞ ¼ 1 if 1 X
PðAn Þ ¼ 1:
n¼1
We first prove Lemma 5.1. The proof is similar to that of Theorem 6 of Adler et al. [1] if we have proved that
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660 1 X ðZ i EZ i Þ
2659
ð17Þ
converges a:s:
i¼1
P P1 whenever Z i is rðX i Þ measurable and 1 i¼1 VarðZ i Þ < 1. Now, Assumption A and i¼1 VarðZ i Þ < 1 imply that 2 j X m X E max ðZ i EZ i Þ 6 C VarðZ i Þ ! 0 as m P n ! 1; n6j6m i¼n i¼n whence 2 j X E max ðZ i EZ i Þ S ! 0 as n ! 1 jPn i¼1 for some random variable S. It follows that j X P max ðZ i EZ i Þ S ! 0: jPn i¼1 Notice the left hand side above is non-increasing in n. We conclude that j X max ðZ i EZ i Þ S ! 0 a:s: jPn i¼1 and (17) is proved. Lemma 5.2 is proved in Zhang and Wen [39]. For completeness of the paper, we give the proof here. If PðAn ; i:o:ðnÞÞ < 1, then there exist n0 large enough and 0 < r < 1 such that ! m [ b ¼: 1 P Ai P r > 0; m P n P n0 : i¼n
By Assumption A and the Cauchy–Schwartz inequality, we have ! ! ( ) m m m m m X X [ X X PðAj Þ ¼ P Aj ; Ai ¼ E ðI Aj PðAj ÞÞ I Sm Ai þ PðAj Þð1 bÞ j¼n
j¼n
i¼n
!
( Var
6
m m [ X ðI Aj Þ P Ai j¼n
( 6
C
m X j¼n
PðAj ÞP
i¼n
j¼n
!)1=2
m X
þ
i¼n m [ i¼n
PðAj Þð1 bÞ
j¼n
!)1=2 Ai
j¼n
þ
m X j¼n
m [ C P PðAj Þð1 bÞ 6 Ai 2b i¼n
! þ
X m b þ1b PðAj Þ; 2 j¼n
whence ! m [ C C PðAj Þ 6 2 P Ai 6 2 ; r b j¼n i¼n P1 which implies j¼1 PðAj Þ < 1. m X
m P n P n0 ;
Acknowledgement The authors thank the editor and anonymous referees for their valuable comments and suggestions, which led to a much improved version of the paper.
2660
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
References [1] A. Adler, A. Rosaksky, R.L. Taylor, Strong laws of large numbers for weighted sums of random elements in normed linear spaces, Int. J. Math. Math. Sci. 12 (3) (1989) 507–530. [2] Z. Artstein, R. Vitale, Strong law of large numbers for random compact sets, Ann. Prob. 3 (1975) 879–882. [3] R. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1C12. [4] R.C. Bradley, On the spectral density and asymptotic normality of weakly dependent random fields, J. Theor. Prob. 5 (1992) 355–373. [5] W. Bryc, W. Smolenski, Moment conditions for almost sure convergence of weakly correlated random variables, Proc. Amer. Math. Soc. 119 (1993) 629–635. [6] A. Colubi, M. Lopez-Diaz, J.S. Dominguez-Menchero, M.A. Gil, A generalized strong law of large numbers, Prob. Theory Related Fields 114 (1999) 401–417. [7] G. Debreu, Integration of correspondence, Proc. Fifth. Berkeley. Symp. Math. Statist. Prob., Univ. of California Press 2 (1952) 351– 372. [8] W. Feller, second ed., An Introduction to Probability Theory and Its Applications, vol. II, Wiley, NewYork, 1971. [9] L. Guan, S.M. Li, Laws of large numbers for weighted sums of fuzzy set-valued random variables, Int. J. Uncertainty Fuzziness Knowledge-based Syst. 12 (2004) 811–825. [10] F. Hiai, H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149–182. [11] H. Inoue, A strong law of large numbers for fuzzy random sets, Fuzzy Sets Syst. 41 (1991) 285–291. [12] S.Y. Joo, Y.K. Kim, J.S. Kwon, Strong convergence for weighted sums of fuzzy random sets, Inform. Sci. 176 (2006) 1086–1099. [13] D.G. Kedall, Foundation of a theory of random set, in: E.F. Harding, D.G. Kendall (Eds.), Stochastic Geometry, Wiley, New York, 1974. [14] E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. London Ser. A 407 (1986) 171– 182. [15] V. Kra¨tschmer, Integrals of random fuzzy sets, Test. 15 (2006) 433–469. [16] R. Kruse, The strong law of large numbers for fuzzy random variables, Inform. Sci. 28 (1982) 233–241. [17] H. Kwakernaak, Fuzzy random variables, Part I: definitions and theorems, Inform. Sci. 15 (1978) 1–29. [18] S.M. Li, Y. Ogura, Fuzzy random variables, conditional expectations and fuzzy valued martingales, J. Fuzzy Math. 4 (1996) 905–927. [19] S.M. Li, Y. Ogura, Strong laws of large numbers for independent fuzzy set-valued random variables, Fuzzy Sets Syst. 157 (2006) 2569–2578. [20] S.M. Li, Y. Ogura, V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Kluwer Academic Publishers Group, Dordrecht, 2002. [21] G. Matherson, Random Set and Integral Geometry, Wiley, New York, 1975. [22] M.L. Puri, D.A. Ralescu, A strong law of large numbers for Banach space-valued random sets, Ann. Prob. 11 (1983) 222–224. [23] M.L. Puri, D.A. Ralescu, Limit theorems for random compact sets in Banach space, Math. Proc. Cambridge Phil. Soc. 97 (1985) 151– 158. [24] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409–422. [25] H. Ra˚dstro¨m, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169. [26] E.E. Robbins, On the measure of a random set, Ann. Math. Stat. 14 (1944) 70–74. [27] E.E. Robbins, On the measure of a random set II, Ann. Math. Stat. 15 (1945) 342–347. [28] E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, vol. 508, Springer-Verlag, Berlin, 1976. [29] R.L. Taylor, Stochastic convergence of weighted sums of random element in linear spaces, Lecture Notes in Mathematics, vol. 672, Springer-Verlag, Berlin, 1978. [30] R.L. Taylor, H. Inoue, Convergence of weighted random sets, Stoch. Anal. Appl. 3 (3) (1985) 379–396. [31] R.L. Taylor, H. Inoue, Law of large numbers for random sets, Random set: theory and applications, vol. 97, Springer, New York, 1997. [32] R.L. Taylor, Y. Seymour, Y. Chen, Strong laws of large numbers for fuzzy random sets, University of Georgia, Department of Statistics, Tech. Report 7, 2000. [33] R.L. Taylor, A.N. Vidyashankar, Y. Chen, Limit theorems for exchangeable random elements and exchangeable random sets, Asymp. Stat. Prob. (2000) 367–378. [34] P. Teran, I. Molchanov, The law of large numbers in a metric space with a convex combination operation, J. Theor. Prob. 19 (2006) 875–898. [35] A. Uemura, A law of large numbers for random sets, Fuzzy Sets Syst. 59 (1993) 181–188. [36] S. Utev, M. Peligrad, Maximal inequalities and an invariance principle for a class of weakly dependent random variables, J. Theor. Prob. 16 (2003) 101–115. [37] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [38] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Inform. Sci. 172 (2005) 1–40. [39] L.X. Zhang, J.W. Wen, Strong limit theorems for B-valued random fields, Chin. Ann. Math. 22A (2001) 205–216 (in Chinese).
Information Sciences 178 (2008) 2648–2660 www.elsevier.com/locate/ins
Strong limit theorems for random sets and fuzzy random sets with slowly varying weights q Ke-ang Fu , Li-xin Zhang Department of Mathematics, Zhejiang University, Hangzhou 310027, China Received 4 November 2005; received in revised form 8 August 2007; accepted 9 January 2008
Abstract Theories of random sets and fuzzy random sets are useful concepts which are frequently applied in scientific areas including information science, probability and statistics. In this paper strong limit theorems are derived for random sets and fuzzy random sets with slowly varying weights in separable Banach spaces. Both independent and dependent cases are covered to provide a wide range of applications. Ó 2008 Elsevier Inc. All rights reserved. AMS classification: 60F15; 60B11 Keywords: Random set; Fuzzy random set; Slowly varying weights; Hausdorff distance; Dependence
1. Introduction In recent years, the theories of random sets and fuzzy random sets have been extensively studied and applied in the areas of information science, probability and statistics. The general idea of random sets has been in existence for some time. Robbins [26,27] appeared to be the first to provide the concept of random sets, and his early works investigated the relationships between random sets and geometric probabilities. Later, Kendall [13] and Matherson [21] provided a comprehensive mathematical theory of random sets which was greatly influenced by the geometric probability prospective. Their proposed framework exerted a strong influence on the limit theorems developed in recent decades. As applications in probability and statistics, statistical inference for random sets leads to the need of laws of large numbers to insure consistency in estimation problems. Hence, laws of large numbers play an important role in statistical inference, and for details one can refer to Artstein and Vitale [2], Puri and Ralescu [22,23], Taylor et al. [29–33], Uemura [35], and references therein. Among them, Artstein and Vitale [2] proved limit theorems concerning random sets in R and Rd ; and Puri and q
Project supported by National Natural Science Foundation of China (Nos. 10671176 & 10771192). Corresponding author. Tel.: +86 13777398392. E-mail addresses: [email protected] (K.-a. Fu), [email protected] (L.-x. Zhang).
0020-0255/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2008.01.005
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
2649
Ralescu [22] were the first to obtain the strong laws of large numbers (SLLN’s) for independent identically distributed (i.i.d.) Banach space-valued compact convex random sets. Among others, SLLN’s were obtained under more relaxed conditions, and a detailed survey of these results is available in Taylor and Inoue [31]. The theory of fuzzy sets was introduced by Zadeh [37] (for an outline recently, one can refer to [38]), and the concept of fuzzy random variables was promoted by Kwakernaak [17] where useful basic properties were developed. Puri and Rasescu [24] used the concept of fuzzy random variables to generalize the results of random sets to fuzzy random sets. With respect to laws of large numbers, Kruse [16] proved an SLLN for i.i.d. fuzzy random variables, and then gave a consistent estimator for the expectation of a fuzzy random variable as an application of the SLLN. Klement et al. [14] considered fuzzy versions of random sets in Euclidean spaces and obtained an i.i.d SLLN. Inoue [11] obtained SLLN’s for independent, tight fuzzy random sets and i.i.d. fuzzy random sets in a separable Banach space. Recently, SLLN’s have been studied under various conditions, and one can refer to the following papers [6,9,12,15,18,19,30,31,34], and references therein. Also for more detailed results about limit theorems of random sets and fuzzy random sets, one can refer to Li et al. [20]. In this article, we establish some strong limit theorems (i.e. strong laws of large numbers) for i.i.d. random sets and fuzzy random sets with slowly varying weights, and we also obtain the sufficient and necessary conditions. The layout of this paper is as follows. In Section 2, we list the basic definitions and properties which will be used to obtain strong limit theorems. In Section 3, we state the main results of this paper and their detailed proofs are given in Section 4. In the last Section we show that our results are also true for a broad class of dependent random sets and fuzzy random sets. In the sequel, let C; C 0 and M denote a positive constant whose value can differ in different places. 2. Definitions and preliminaries Throughout this paper, let S be a real separable Banach space with the norm k k and the dual space S . For each A S; clA and coA denote the norm-closure and the closed convex hull of A, respectively. Let CðSÞ (resp. C c ðSÞ) denote the collections of all non-empty compact (resp. non-empty closed compact) subsets of S. Define the Minkowski’s addition and scalar multiplication, respectively, in CðSÞ (or C c ðSÞ) by A þ B ¼ fa þ bja 2 A; b 2 Bg; kA ¼ fkaja 2 Ag; where A; B 2 CðSÞ (or C c ðSÞ) and k is a real number. Neither CðSÞ nor C c ðSÞ are linear spaces even when S ¼ R, one-dimensional Euclidean space. And for A; B 2 CðSÞ, the Hausdorff distance d H ðA; BÞ of A and B and the norm kAk of A are defined by d H ðA; BÞ ¼ maxfsup inf ka bk; sup inf ka bkg; a2A b2B
b2B a2A
kAk ¼ d H ðA; f0gÞ ¼ sup kak: a2A
Let ðX; F; PÞ denote a probability measure space. A random (compact) set is a Borel measurable function F : X ! CðSÞ; that is, F 1 ðBÞ ¼ fx 2 X; F ðxÞ \ B 6¼ ;g 2 F for each B 2 CðSÞ (c.f. [10,20]). For a random set F in CðSÞ; there exists a corresponding set coF in C c ðSÞ; which can be used in defining an expected value. A measurable function f : X ! S is called a measurable selection of F if f ðxÞ 2 F ðxÞ for every x 2 X. Denote by S F ¼ ff 2 L1 ðX; SÞ; f ðxÞ 2 F ðxÞ; a:e:g;
R where L1 ðX; SÞ denotes the space of measurable functions f : X ! S such that n kf ðxÞkdP < 1. S F 6¼ ; if and only if the random variable kF ðxÞk is integrable. The random set F is called integrably bounded if the realvalued random variable kF ðxÞk is integrable (c.f. [10,20]). Hiai and Umegaki [10] showed that a random set F is integrably bounded if and only if S F is bounded in L1 ðX; SÞ. Thus an integrably bounded random set may take unbounded sets.
2650
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
For each random set F, the expectation of F, denoted by EðF Þ; is defined by Z Z EðF Þ ¼ F dP ¼ f dP; f 2 S F ; X
ð1Þ
X
R R R where X f dP is the usual Bochner integral in L1 ðX; SÞ. Define A F dP ¼ A f dP; f 2 S F for A 2 F. This definition was introduced by Aumann in 1965 as a natural generalization of the integral of real-valued random variables Rin [3]. For the random set F, if EkcoF k < 1; then a Bochner integral can be defined as EðcoF Þ ¼ X coF dP and EðcoF Þ 2 C c ðSÞ [7]. For A, B, C and D 2 CðSÞ; k 2 R and X ; Y are random sets, immediate properties of the Hausdorff distance lead to the following: d H ðkA; kBÞ 6 jkjd H ðA; BÞ; d H ðA; BÞ 6 d H ðA; CÞ þ d H ðC; BÞ; d H ðA þ C; B þ DÞ 6 d H ðA; BÞ þ d H ðC; DÞ; d H ðcoA; coBÞ 6 d H ðA; BÞ; d H ðEcoX ; EcoY Þ 6 Ed H ðX ; Y Þ: A finite set of random sets fX 1 ; . . . ; X n g in CðSÞ is said to be independent if PðX 1 2 B1 ; . . . ; X n 2 Bn Þ ¼ PðX 1 2 B1 Þ PðX n 2 Bn Þ for every B1 ; . . . ; Bn 2 CðSÞ. A family of random set in CðSÞ is said to be independent if every finite subset is independent. The random sets X and Y are said to be identical distributed if PðX 2 BÞ ¼ PðY 2 BÞ for every B 2 CðSÞ. A family of random sets is identically distributed if every pair is identically distributed. From the definitions above, we know that most stochastic properties of random sets are extended from real-valued random variables directly. Now we begin to state some properties about fuzzy sets. A fuzzy set in S is a function u : S ! ½0; 1. Let F ðSÞ denote the family of the fuzzy subset u satisfying the following conditions: (a) u is upper semicontinuous, that is, the a-level set of u, i.e. ua ¼ fx 2 S; uðxÞ P ag is a closed subset of S for each a 2 ð0; 1, (b) fx 2 S : uðxÞ > 0g has compact closure, (c) fx 2 S : uðxÞ ¼ 1g 6¼ ;. For u 2 F ðSÞ; the support of u is defined as supp u ¼ fx 2 S : uðxÞ > 0g, and the assumption of compact closure implies that support of u is norm bounded. A linear structure in F ðSÞ is defined by the following operations: ðu þ vÞðxÞ ¼ sup min½uðyÞ; vðzÞ; yþz¼x 1 uðk xÞ; if k 6¼ 0; ðkuÞðxÞ I 0 ðxÞ; if k ¼ 0; where u; v 2 F ðSÞ; k 2 R; and I 0 ðÞ is an indicator function. The linear structure aforementioned implies the following properties: ðu þ vÞa ¼ ua þ va ; ðkuÞa ¼ kua : And for fuzzy sets, we adopt the most common metric d r , viewed as a generalization of the Hausdorff metric from CðSÞ to F ðSÞ; where Z 1 1=r d r ðu; vÞ ¼ d rH ðua ; va Þda ; ð2Þ 0
where u; v 2 F ðSÞ and 1 6 r 6 2. Also we define a topology T using the countable family of semimetrics d~a ðu; vÞ ¼ d H ðua ; va Þ;
a2 W;
where W denotes the dyadic rational numbers, i.e. W ¼ f2im : 1 6 i 6 2m ; m P 1g.
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
2651
Definition 2.1. The topology T is defined by the smallest which makes each projection La , a 2 W ; P 1topology d~ai ðu;vÞ continuous. The topology is meterizable by d T ¼ 1 for all u; v 2 F ðSÞ; where a1 ; a2 ; . . . is a i i¼1 2 1þd~a ðu;vÞ i countable listing of W. Since every semimetric d~a is separable for F ðSÞ; it follows that T is separable. Taylor et al. [32] showed that T and d r generate the same Borel subsets, and hence have the same fuzzy random sets defined as below. The concept of a fuzzy random set as a generation for a random set was extensively studied by Puri and Ralescu [24]. A fuzzy random set is a function X : X ! F ðSÞ such that for each a 2 ð0; 1, X a ðxÞ ¼ fx 2 S; X ðxÞðxÞ P ag is a random set in S (c.f. [20]). A random fuzzy set X is said to be integrably bounded if the real-valued random variable ksupp uk is integrable. The expectation of a fuzzy random set X, denoted by E½X ; is an element in F ðSÞ such that for each a 2 ð0; 1; Z ðE½X Þa ¼ cl X a dP ¼ clfEðf Þ; f 2 S X a g; X
where the closure is taken in S and S X a ¼ ff 2 L1 ðX; SÞ; f ðxÞ 2 X a ðxÞ a:e:g. By virtue of the existence theorem (c.f. [20]), we have an equivalent definition as follows: E½X ðxÞ ¼ supfa 2 ð0; 1; x 2 E½X a g: Furthermore, ðE½coX Þa ¼ E½ðcoX Þa for any a 2 ð0; 1. A sequence fuzzy random sets fX n ; n P 1g is said to be independent if for any a 2 ð0; 1; the sequence of random sets fX n a; n P 1g is independent, and the identical distribution can be defined similarly. Now we introduce something about slowly varying functions. Definition 2.2. A monotone (increasing or decreasing) function L : ½a; 1Þ ! ð0; 1Þ, where a P 0, is said to be slowly varying if lim
x!1
LðtxÞ ¼1 LðxÞ
8t > 0:
ð3Þ
It is well known [28] that if L is slowly varying, then limx!1 Lðx þ 1Þ=LðxÞ ¼ 1 and LðxÞ ¼ Oðxi Þ and 1=LðxÞ ¼ Oðxi Þ as x ! 1; for any i > 0:
ð4Þ
Also P if L is a slowly varying function defined on ½1; 1Þ, then it follows from (4) and Theorem 1(b) of Feller [8] n that j¼1 LðjÞ nLðnÞ % 1. 3. Main results In this section, we list the strong laws of large numbers for random sets and fuzzy random sets, respectively. The first one is for independent and identical distributed (i.i.d.) random sets, and we establish the sufficient and necessary conditions. Theorem 3.1. Let fX n ; n P 1g be a sequence of i.i.d. random sets in CðSÞ, (i) let L be a slowly varying function defined on ½1; 1Þ, and let m 2 C c ðSÞ, then Pn j¼1 LðjÞcoX j ! m a:s: ð5Þ Rn1 :¼ nLðnÞ if and only if EðkcoX 1 kÞ < 1
and
EcoX 1 ¼ m;
ð6Þ
(ii) Moreover, Pn Rn2 :¼
j¼1 LðjÞX j
nLðnÞ
!m
a:s:
ð7Þ
if and only if (6) holds, where the convergence is in the Hausdorff metric d H of CðSÞ and co denotes the continuous mapping of CðSÞ into C c ðSÞ.
2652
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
Remark 3.1. If the identical distribution hypothesis is relaxed to that the random sets are (uniformly) tight and stochastically dominated by a random variable X, then the conclusions above remain true. Remark 3.2. Taking LðjÞ ¼ C 0 (a fixed constant) yields the following Corollary which has been proved with various approaches. Corollary 3.1. Let fX n ; n P 1g be a sequence of i.i.d. random sets in CðSÞ, such that EðkcoX 1 kÞ < 1 and EðcoX 1 Þ ¼ m. Then in Hausdorff sense n 1X X j ! m a:s: ð8Þ n j¼1 The following result is for i.i.d. fuzzy random sets, but we only give the sufficient condition. Theorem 3.2. Let fX n ; n P 1g be a sequence of i.i.d. fuzzy random sets in F ðSÞ such that EkX 1 kr < 1, and let Rn2 be defined in form of (7). Then we have d r ðRn2 ; EX 1 Þ ! 0 a:s:;
ð9Þ
where r is defined by the metric d r in (2). r
Conjecture 3.1. We believe that EkX 1 k < 1 is still the necessary condition for fuzzy random sets. To get such an improvement of the results, we think a different approach is necessary. 4. Proofs In this section, we will give the detailed proof for the two theorems above. And to prove the theorems, we need to introduce the following lemmas. Lemma 4.1 [1]. Let fX n ; n P 1g be real independent random variables and let X be a real random variable with EjX jp < 1 for some 1 6 p < 2. Suppose that fX n ; n P 1g is stochastically dominated by X in the sense that there exits a constant D < 1 such that PðjX n j > tÞ 6 DPðjDX j > tÞ;
t P 0; n P 1: 1
Let fan ; n P 1g and fbn ; n P 1g be constants satisfying 0 < bn " 1, an =bn ¼ Oðnp Þ and we have the SLLN: n X aj ðX j EX j Þ ! 0 a:s: b1 n
Pn
j¼1 jaj j
¼ Oðbn Þ. Then
j¼1
Lemma 4.2 [1]. Let X 0 and X be real random variables such that X 0 is stochastically dominated by X in the sense that PðjX 0 j > tÞ 6 DPðjDX j > tÞ holds for all t > 0. Then Z 1 EjX 0 jIðjX 0 j > xÞ ¼ PðjX 0 j > tÞdt þ xPðjX 0 j > xÞ; x P 0 x
and EjX 0 jIðjX 0 j > xÞ 6 D2 EjX jIðjDX j > xÞ;
x P 0:
The following two lemmas are only slight modifications of the proofs in [1]. They are stated here for the convenience of the readers. Lemma 4.3. Let fV n ; n P 1g be independent, mean zero random elements in a separable Banach space admitting a Schauder basis fbi g with the coordinate functional ffi g, and let fan ; n P 1g and fbn ; n P 1g be defined as those
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
2653
P P1 in Lemma 4.1. Denote U m ðV Þ ¼ m i¼1 fi ðV Þbi and Qm ðV Þ ¼ i¼mþ1 fi ðV Þbi ; m P 1. Suppose that there exist real random variables fX i ; i P 1g and fY m ; m P 1g and a constant D < 1 such that for some 1 6 p < 2, Pðjfi ðV n Þj > tÞ 6 DPðjDX i j > tÞ;
t P 0; n P 1; i P 1;
PðjkQm ðV n Þ EkQm ðV n Þkj > tÞ 6 DPðjDY m j > tÞ; p
t P 0; n P 1; m P 1;
p
sup EjX i j < 1;
sup EjY m j < 1
iP1
mP1
and lim sup EkQm ðV n Þk ¼ 0:
ð10Þ
m!1 nP1
Then we have the SLLN n X aj V j ! 0 a:s b1 n j¼1
. Proof. It follows immediately from Lemma 4.1 that n X b1 aj fi ðV j Þ ! 0 a:s: for each i P 1 n
ð11Þ
j¼1
and T mn :¼ b1 n
n X
jaj jðkQm ðV j Þk EkQm ðV j ÞkÞ ! 0
a:s: for each m P 1:
ð12Þ
j¼1
Also it follows that ! ! X X n m n m n X X 1 X 1 1 aj V j ¼ fi bn aj V j b i 6 aj fi ðV j Þ kbi k: U m b n bn j¼1 i¼1 j¼1 i¼1 j¼1 Pn Thus, by (11)–(13) and j¼1 jaj j ¼ Oðbn Þ, we have ! ! X n n n X X b1 aj V j 6 U m b1 aj V j þ Qm b1 aj V j n n n j¼1 j¼1 j¼1 ! X n n X aj V j þ b1 aj Qm ðV j Þ 6 U m b1 n n j¼1 j¼1 ! n X 6 U m b1 aj V j þ T mn þ C sup EkQm ðV j Þk ! 0 a:s: n jP1 j¼1
ð13Þ
as m ! 1 after letting n ! 1. h Lemma 4.4. Let fV n ; n P 1g be a (uniformly) tight sequence of independent, mean zero random elements in a Banach space E, and let fan ; n P 1g and fbn ; n P 1g be defined as those in Lemma 4.1. Suppose that p fV nP ; n P 1g is stochastically dominated by a random element V and EkV k < 1 for some 1 6 p < 2. Then n 1 bn j¼1 aj V j ! 0 a.s. Proof. Since the isometrically embedding of E into a Banach space which has a monotone Schauder basis preserves the independence and uniformly tightness, we might as well assume that E has a monotone basis where kQm ðyÞk 6 kyk and kfm ðyÞk 6 kyk for each y 2 E and m P 1 and fkQm ðyÞk; m P 1g is a monotone decreasing sequence for each y 2 E. Now, for 8e > 0, choose u > 0 such that D2 EkV kIðkV k > uÞ 6 e=3:
2654
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
Thus, from Lemma 4.2, it follows EkV n kIðkV n k > uÞ 6 e=3 for all n P 1. By (uniformly) tightness, we can choose a compact subset K such that PðV n 62 KÞ 6 e=ð3uÞ. Thus it yields EkV n kIðkV n k 6 uÞIðV n 62 KÞ 6 e=3 for all n P 1. Since kQm ðzÞk # 0 for all z 2 K, there exists an integer m0 such that supz2K kQm ðzÞk 6 e=3 for all m P m0 . Hence, for all m P m0 and n P 1, EkQm ðV n Þk 6 EkQm ðV n ÞIðkV n k 6 uÞIðV n 2 KÞk þ EkV n kIðkV n k 6 uÞIðV n 62 KÞ þ EkV n kIðkV n k > uÞ 6 e: Since e is arbitrary, we get that limm!1 supnP1 EkQm ðV n Þk ¼ 0. Now, Pn let X i ¼ kV i k, and Y m ¼ kV m k þ DEkV m k for all i P 1; m P 1. Applying Lemma 4.3, we have that b1 n k j¼1 aj V j k ! 0 a:s. h Remark 4.1. From the proof above, we can obtain the following conclusion which will be used later: for 8e > 0, we can find a compact subset K such that EkV n IðV n 62 KÞk 6 e:
ð14Þ
If fV n ; n P 1g is a sequence of identically distributed random elements, the condition of stochastic domination is automatically satisfied with V ¼ V 1 and D ¼ 1. Furthermore, identical distribution property ensures that fV n ; n P 1g is automatically (uniformly) tight [29]. Hence, we have the following proposition. Proposition 4.5. Let fV n ; n P 1g be a sequence of i.i.d. random elements in a separable Banach space E, and let p fan ; nPP 1g and fbn ; n P 1g be defined as that in Lemma 4.1. If EkV n k < 1 for some 1 6 p < 2, then n 1 bn k j¼1 aj ðV j EV j Þk ! 0 a:s. Ra˚dstro¨m [25] showed that the collection of compact convex subsets of a Banach space can be embedded as a convex cone in a normed linear space. That is, the metric space C c ðSÞ is embedded in a separable Banach space N with an isometry g : C c ðSÞ ! N. Lemma 4.6 [25]. Let X : X ! C c ðSÞ be a random set such that EkX k < 1. If g : C c ðSÞ ! N is the isometry given by the Ra˚dstro¨P m embedding P theorem, then Eðg X Þ ¼ gðEX Þ. In particular, if xi 2 C c ðSÞ, ai P 0, i ¼ 1; 2; . . . ; n, then gð ni¼1 ai xi Þ ¼ ni¼1 ai gðxi Þ. Lemma 4.7 [30]. Let A 2 CðSÞ and fank g be an array of non-negative constants such that n X ank 6 1; ðiiÞ max ank ! 0 as n ! 1: ðiÞ k¼1
Then d H ð
Pn
k¼1 ank ck A;
16k6n
Pn
k¼1 ank ck coAÞ
! 0 for any sequence fcn g consisting of 1’s and 0’s.
Now, we state the proof of Theorem 3.1. Proof of Theorem 3.1. (i) Sufficiency. Assume that (6) holds and let g be defined as that in Lemma 4.6. Let an ¼ LðnÞ and P bn ¼ nj¼1 LðjÞ; n P 1. Then bn " 1. Also we know that bn nLðnÞ and abnn ¼ Oð1nÞ from preliminaries. By the distance preserving property of g and Lemma 4.6, ! ! ! n n n n X X X X aj coX j ; b1 aj EcoX j ¼ g b1 aj coX j g b1 aj EcoX j d H b1 n n n n j¼1 j¼1 j¼1 j¼1 n X
ð15Þ aj gðcoX j Þ EgðcoX j Þ : ¼ b1 n j¼1 Note that g coX ¼ gðcoX Þ is a random element in Banach space N, and EkgðcoX Þk ¼ EkcoX k < 1. Thus it follows from Proposition 4.5 that
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
n 1 X
aj gðcoX j Þ EgðcoX j Þ ! 0 bn j¼1
2655
a:s:
Hence, by applying (15), we have ! ! ! n n n n X X X X 1 1 1 1 d H bn LðjÞcoX j ; m ¼ d H bn LðjÞcoX j ; EcoX 1 ¼ d H bn aj coX j ; bn aj EcoX j j¼1
j¼1
!0
j¼1
j¼1
a:s:
Pn LðjÞcoX j ! m a.s. with respect to d H . Since bn nLðnÞ, we can easily get that Rn1 ¼ j¼1nLðnÞ Necessity. Assume that (5) holds. Then we have Pn Pn Pn
j¼1 LðjÞgðcoX j Þ j¼1 LðjÞcoX j j¼1 LðjÞcoX j gðmÞ ¼ g ;m ! 0 gðmÞ ¼ d H nLðnÞ nLðnÞ nLðnÞ Thus it follows that Pn Pn1 gðcoX n Þ ðn 1ÞLðn 1Þ j¼1 LðjÞgðcoX j Þ j¼1 LðjÞgðcoX j Þ ¼ ! gðmÞ 1 gðmÞ ¼ 0 n nLðnÞ nLðnÞ ðn 1ÞLðn 1Þ
a:s:
a:s:
in the sense that kcoX n k=n ¼ kgðcoX n Þk=n ! 0 a:s. Hence, we have PðkcoX n k=n > 1; i:o:ðnÞÞ ¼ 0. And coupled with the i.i.d. hypothesis and the Borel–Cantelli lemma, it leads to 1 1 X X PðkcoX 1 k > nÞ ¼ PðkcoX n k > nÞ < 1: n¼1
n¼1
Then it follows EkcoX n k ¼ EkcoX 1 k < 1; i.e., EcoX 1 exists. By the sufficiency partition of the theorem, we have Rn1 ! EcoX 1 a:s:; and hence together with (4) it implies EcoX 1 ¼ m. Thus the proof of part (i) is completed. (ii) Sufficiency. Since fX n ; n P 1g is i.i.d., as explained in Remark 4, for any e > 0, one can choose a compact subset K of CðSÞ such that EkX n IðX n 62 KÞk ¼ EkX 1 IðX 1 62 KÞk 6 e for all n, and since K is compact, there exist k 1 ; . . . ; k m 2 K such that m m [ [ Bðk j ; eÞ: K fz : kz k j k < eg
j¼1
j¼1
Z 0n IðX n
2 KÞ be a random set taking finitely many values, where Z 0n ¼ k 1 I½X n 2 Bðk 1 ; eÞþ Let Pm Z n ¼ Si1 c i¼2 k i IfðX Pnn 2 Bðk i ; eÞÞ \ l¼1 ðX n 2 Bðk l ; eÞ Þg. Thus fZ n ; n P 1g is also a sequence of i.i.d. random sets. Let bn ¼ j¼1 LðjÞ; and then we have Pn Pn
Pn
j¼1 LðjÞX j j¼1 LðjÞX j j¼1 LðjÞEcoX 1 ; m ¼ dH ; dH bn bn bn Pn Pn
j¼1 LðjÞX j j¼1 LðjÞX j IðX j 2 KÞ 6 dH ; bn bn P Pn Pn n
Pn
j¼1 LðjÞX j IðX j 2 KÞ j¼1 LðjÞZ j j¼1 LðjÞZ j j¼1 LðjÞEcoZ 1 þ dH ; ; þ dH bn bn bn bn þ d H ðEcoZ 1 ; EcoX 1 IðX 1 2 KÞÞ þ d H ðEcoX 1 IðX 1 2 KÞ; EcoX 1 Þ ¼: ðIÞ þ ðIIÞ þ ðIIIÞ þ ðIVÞ þ ðVÞ: Now we begin to deal with (I)–(V), respectively. For (I), Pn j¼1 LðjÞ kX j IðX j 62 KÞk: ðIÞ 6 bn
2656
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
Since fkX n IðX n 62 KÞk EkX n IðX n 62 KÞkg is a sequence of i.i.d. random variables with zero means, by Lemma 4.1 we can get that
Pn j¼1 LðjÞ kX j IðX j 62 KÞk EkX j IðX j 62 KÞk ! 0 a:s: bn Thus it follows that n X ðIÞ 6 b1 LðjÞkX j IðX j 62 KÞk ! EkX 1 IðX 1 62 KÞk 6 e a:s: n j¼1
As to (II), by the construction of Z n ; it implies n X LðjÞd H ðX j IðX j 2 KÞ; Z j Þ 6 e: ðIIÞ 6 b1 n j¼1
With regard to (III), we have Pn Pn
Pn
j¼1 LðjÞZ j j¼1 LðjÞcoZ j j¼1 LðjÞcoZ j ðIIIÞ 6 d H ; ; EcoZ 1 ¼: ðIII1 Þ þ ðIII2 Þ: þ dH bn bn bn Here we use Lemma 4.7 for cj ¼ IðZ j ¼ k i Þ and A ¼ k i . Since the number m is finite, we then obtain Pm Pm Pn Pn
j¼1 LðjÞ i¼1 k i IðZ j ¼ k i Þ j¼1 LðjÞ i¼1 cok i IðZ j ¼ k i Þ ðIII1 Þ ¼ d H ; bn bn Pn Pn
m X j¼1 LðjÞk i IðZ j ¼ k i Þ j¼1 LðjÞcok i IðZ j ¼ k i Þ 6 dH ; ! 0 a:s: bn bn i¼1 And ðIII2 Þ follows directly from the first part of Theorem 3.1. For (IV), from the construction of Z 1 and preliminaries, it follows that d H ðEcoZ 1 ; EcoX 1 IðX 1 2 KÞÞ 6 Ed H ðZ 1 ; X 1 IðX 1 2 KÞÞ 6 e: And about (V), we have d H ðEcoX 1 IðX 1 2 KÞ; EcoX 1 Þ 6 Ed H ðX 1 IðX 1 2 KÞ; EX 1 Þ ¼ EkX 1 IðX 1 62 KÞk 6 e: Thus by the arbitrariness of e, we have that ! n X 1 d H bn LðjÞX j ; m ! 0 a:s: j¼1 Pn LðjÞX j and hence, it follows that Rn2 ¼ j¼1 ! m a:s. with respect to d H due to bn nLðnÞ. nLðnÞ Necessity. Assume that (7) holds. Then we have Pn j¼1 LðjÞX j nLðnÞ ! kmk a:s: Notice that kAk 6 kA þ Bk þ kBk for sets A and B. It follows that # Pn1 "Pn j¼1 LðjÞX j ðn 1ÞLðn 1Þ LðjÞX j kX n k j¼1 þ lim sup 6 lim ¼ 2kmk a:s: n!1 n nLðnÞ nLðnÞ ðn 1ÞLðn 1Þ n!1 Since Pðkmk 6 MÞ > 0 for some constant M, we conclude that P kXnn k > 3M; i:o:ðnÞ < 1. By the i.i.d. hypothesis and the Borel–Cantelli lemma, we have 1 1 X X PðkX 1 k > 3MnÞ ¼ PðkX n k > 3MnÞ < 1: n¼1
n¼1
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
2657
Thus it follows that EkcoX 1 k 6 EkX 1 k < 1, i.e., EcoX 1 exists. By the sufficiency partition of the theorem, it leads to Rn2 ! EcoX 1 a:s. Hence, we have EcoX 1 ¼ m via (7). The proof is now terminated. Before proving Theorem 3.2, we need two more lemmas related to fuzzy random sets. Lemma 4.8 [32]. If supn ksupp un k < 1 and un ! u in T, then we have that for 1 6 r 6 2 d r ðun ; uÞ ! 0
as n ! 1:
Lemma 4.9 [32]. Let X be a fuzzy random set in F ðSÞ such that EkX krr < 1. Then we have Z
1
lim
M!1
EkX aIðkX ak > MÞkr da ¼ 0;
r P 1:
0
Now we turn to the Proof of Theorem 3.2. For every e > 0, choose an M > 0 such that Z 1 r EkX 1 aIðkX 1 ak > MÞk da < er 0
and for P every a 2 ½0; P1, set Ak a ¼ X k aIðkX k ak 6 MÞ and Bk a ¼ X k aIðkX k ak > MÞ. Note that X k a ¼ Ak a þ Bk a and ð nk¼1 X k Þa ¼ nk¼1 ðX k aÞ. Thus it follows that X k a0 s, Ak a0 s and Bk a0 s are all i.i.d. random sets due to the fact that X 0k s are i.i.d. And hence we have Z 1 1=r Z 1 Pn
1=r j¼1 LðjÞX j a r ; EcoðX 1 aÞ da d r ðRn2 ; EX 1 Þ ¼ d H ðRn2 a; EðX 1 aÞÞda ¼ d rH nLðnÞ 0 0 Z 1 Pn
1=r j¼1 LðjÞðAj a þ Bj aÞ ; EcoðA1 a þ B1 aÞ da ¼ d rH nLðnÞ 0 Z 1 Pn
1=r Z 1 Pn
1=r j¼1 LðjÞAj a j¼1 LðjÞBj a r r 0 0 ; EcoðA1 aÞ da ; EcoðB1 aÞ da 6C dH þC dH nLðnÞ nLðnÞ 0 0 ¼: C 0 ððVIÞ þ ðVIIÞÞ: For (VI), since kAk ak 6 M, Theorem 3.1 provides that for every a 2 W ; Pn
r j¼1 LðjÞAj a ; EcoðA1 aÞ dH ! 0 a:s: as n ! 1: nLðnÞ Since the dyadic rationales W are countable and fAk ag are bounded by M, it follows from Lemma 4.8 that
Pn
Z 1 Pn Z 1 j¼1 LðjÞAj a j¼1 LðjÞAj a r r ; EcoðA1 aÞ da ¼ ; EcoðA1 aÞ da ¼ 0 a:s: lim dH lim d H ð16Þ n!1 0 nLðnÞ nLðnÞ 0 n!1 For (VII), we have Z
1
d rH
0
R1
Pn
j¼1 LðjÞBj a
nLðnÞ
1=r
1=r (Z ; EcoðB1 aÞ da 6
Pn )1=r Z 1 1=r j¼1 LðjÞBj ar r da þ EkB1 ak da nLðnÞ 0 0 Z 1
1=r Xn 1 r 6 ðnLðnÞÞ LðjÞ kBj ak da þ e: j¼1
1
0
Since kBj akr da ;j P 1 is a sequence of i.i.d. random variables with Eð 0 R1 r r EkB1 ak da < e , by Lemma 4.1, we have 0
R1 0
kB1 akr daÞ ¼
2658
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
ðnLðnÞÞ
1
n X
Z LðjÞ
1 r
1=r
kBj ak da
!E
Z
0
j¼1
1 r
kB1 ak da
1=r
Z
1 r
EkB1 ak da
6
0
1=r <e
a:s:
0
Consequently, we obtain the desired relationship (9) by letting e ! 0. 5. Dependent random sets and fuzzy random sets In this section we show that our main results can be generalized to the case of dependent random sets and fuzzy random sets. When the sequence is not independent, we assume the following assumption instead. Assumption A. Let fX n ; n P 1g be a sequence of random variables (random elements, random sets or fuzzy random sets). Suppose for any sequence fY i ; i P 1g of real random variables with Y i being rðX i Þ measurable and EjY i j2 < 1, there exists a constant C > 0 such that 2 j X m X E max ðY i EY i Þ 6 C VarðY i Þ; m P n P 1: n6j6m i¼n i¼n When X 1 ; X 2 ; . . . are independent, Assumption A is obviously satisfied with C ¼ 2. This Assumption is also satisfied for many dependent random variables. For example, if fX n ; n P 1g is a u-mixing sequence wimixing P coefficient satisfying i uðiÞ < 1, Assumption A is satisfied. Mixing sequences are very popular in modeling dependent stochastic phenomena. Recently, Bradley [4], Byrc and Smolenjki [5], Utev and Peligrad [36] etc. studied a particular type of weakly mixing random variables defined as follows. For a random sequence fX n ; n P 1g, denote FT ¼ rðX i ; i 2 T Þ where T is a family of positive integers. Define qn ¼ sup
sup
jCorrðf ; gÞj;
f 2L2 ðFS Þ;g2L2 ðFT Þ
where the first sup is taken over all pairs of non-empty finite sets S, T of positive integers such that distðS; T Þ P n. Obviously, 0 6 qn 6 1. Many limit theorems are obtained for real random variables under the weak dependence condition that limn!1 qn < 1 (c.f., [4,5,36]). This weak dependence condition means that the random variables are not highly correlated when their distance is large enough. According to Theorem 2.1 of Utev and Peligrad [36], a random sequence fX n ; n P 1g satisfies Assumption A if limn!1 qn < 1. Theorem 5.1. Theorems 3.1 and 3.2 remain true with the conditions unchanged except that the condition X 1 ; X 2 ; are independent is replaced by Assumption A. The proof is similar to the one of Theorems 3.1 and 3.2 if Lemma 4.1 and the converse part of the Borel– Cantelli lemma are replaced by Lemmas 5.1 and 5.2 below, respectively. Lemma 5.1. Suppose Assumption A is satisfied and Y, Y 1 ; Y 2 ; . . . are random variables such that Y i is rðX i Þ measurable and EjY jp < 1 for some 1 6 p < 2. Assume that fY n ; n P 1g is stochastically dominated by1 Y. Let fan ; n P 1g and fbn ; n P 1g be sequences of constants satisfying 0 < bn " 1, an =bn ¼ Oðnp Þ and P n j¼1 jaj j ¼ Oðbn Þ. Then we have the SLLN: n X aj ðY j EY j Þ ! 0 a:s: b1 n j¼1
Lemma 5.2. Suppose Assumption A is satisfied. Let fAi ; i P 1g be a sequence of events such that Ai 2 rðX i Þ. Then PðAn ; i:o:ðnÞÞ ¼ 1 if 1 X
PðAn Þ ¼ 1:
n¼1
We first prove Lemma 5.1. The proof is similar to that of Theorem 6 of Adler et al. [1] if we have proved that
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660 1 X ðZ i EZ i Þ
2659
ð17Þ
converges a:s:
i¼1
P P1 whenever Z i is rðX i Þ measurable and 1 i¼1 VarðZ i Þ < 1. Now, Assumption A and i¼1 VarðZ i Þ < 1 imply that 2 j X m X E max ðZ i EZ i Þ 6 C VarðZ i Þ ! 0 as m P n ! 1; n6j6m i¼n i¼n whence 2 j X E max ðZ i EZ i Þ S ! 0 as n ! 1 jPn i¼1 for some random variable S. It follows that j X P max ðZ i EZ i Þ S ! 0: jPn i¼1 Notice the left hand side above is non-increasing in n. We conclude that j X max ðZ i EZ i Þ S ! 0 a:s: jPn i¼1 and (17) is proved. Lemma 5.2 is proved in Zhang and Wen [39]. For completeness of the paper, we give the proof here. If PðAn ; i:o:ðnÞÞ < 1, then there exist n0 large enough and 0 < r < 1 such that ! m [ b ¼: 1 P Ai P r > 0; m P n P n0 : i¼n
By Assumption A and the Cauchy–Schwartz inequality, we have ! ! ( ) m m m m m X X [ X X PðAj Þ ¼ P Aj ; Ai ¼ E ðI Aj PðAj ÞÞ I Sm Ai þ PðAj Þð1 bÞ j¼n
j¼n
i¼n
!
( Var
6
m m [ X ðI Aj Þ P Ai j¼n
( 6
C
m X j¼n
PðAj ÞP
i¼n
j¼n
!)1=2
m X
þ
i¼n m [ i¼n
PðAj Þð1 bÞ
j¼n
!)1=2 Ai
j¼n
þ
m X j¼n
m [ C P PðAj Þð1 bÞ 6 Ai 2b i¼n
! þ
X m b þ1b PðAj Þ; 2 j¼n
whence ! m [ C C PðAj Þ 6 2 P Ai 6 2 ; r b j¼n i¼n P1 which implies j¼1 PðAj Þ < 1. m X
m P n P n0 ;
Acknowledgement The authors thank the editor and anonymous referees for their valuable comments and suggestions, which led to a much improved version of the paper.
2660
K.-a. Fu, L.-x. Zhang / Information Sciences 178 (2008) 2648–2660
References [1] A. Adler, A. Rosaksky, R.L. Taylor, Strong laws of large numbers for weighted sums of random elements in normed linear spaces, Int. J. Math. Math. Sci. 12 (3) (1989) 507–530. [2] Z. Artstein, R. Vitale, Strong law of large numbers for random compact sets, Ann. Prob. 3 (1975) 879–882. [3] R. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1C12. [4] R.C. Bradley, On the spectral density and asymptotic normality of weakly dependent random fields, J. Theor. Prob. 5 (1992) 355–373. [5] W. Bryc, W. Smolenski, Moment conditions for almost sure convergence of weakly correlated random variables, Proc. Amer. Math. Soc. 119 (1993) 629–635. [6] A. Colubi, M. Lopez-Diaz, J.S. Dominguez-Menchero, M.A. Gil, A generalized strong law of large numbers, Prob. Theory Related Fields 114 (1999) 401–417. [7] G. Debreu, Integration of correspondence, Proc. Fifth. Berkeley. Symp. Math. Statist. Prob., Univ. of California Press 2 (1952) 351– 372. [8] W. Feller, second ed., An Introduction to Probability Theory and Its Applications, vol. II, Wiley, NewYork, 1971. [9] L. Guan, S.M. Li, Laws of large numbers for weighted sums of fuzzy set-valued random variables, Int. J. Uncertainty Fuzziness Knowledge-based Syst. 12 (2004) 811–825. [10] F. Hiai, H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149–182. [11] H. Inoue, A strong law of large numbers for fuzzy random sets, Fuzzy Sets Syst. 41 (1991) 285–291. [12] S.Y. Joo, Y.K. Kim, J.S. Kwon, Strong convergence for weighted sums of fuzzy random sets, Inform. Sci. 176 (2006) 1086–1099. [13] D.G. Kedall, Foundation of a theory of random set, in: E.F. Harding, D.G. Kendall (Eds.), Stochastic Geometry, Wiley, New York, 1974. [14] E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. London Ser. A 407 (1986) 171– 182. [15] V. Kra¨tschmer, Integrals of random fuzzy sets, Test. 15 (2006) 433–469. [16] R. Kruse, The strong law of large numbers for fuzzy random variables, Inform. Sci. 28 (1982) 233–241. [17] H. Kwakernaak, Fuzzy random variables, Part I: definitions and theorems, Inform. Sci. 15 (1978) 1–29. [18] S.M. Li, Y. Ogura, Fuzzy random variables, conditional expectations and fuzzy valued martingales, J. Fuzzy Math. 4 (1996) 905–927. [19] S.M. Li, Y. Ogura, Strong laws of large numbers for independent fuzzy set-valued random variables, Fuzzy Sets Syst. 157 (2006) 2569–2578. [20] S.M. Li, Y. Ogura, V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Kluwer Academic Publishers Group, Dordrecht, 2002. [21] G. Matherson, Random Set and Integral Geometry, Wiley, New York, 1975. [22] M.L. Puri, D.A. Ralescu, A strong law of large numbers for Banach space-valued random sets, Ann. Prob. 11 (1983) 222–224. [23] M.L. Puri, D.A. Ralescu, Limit theorems for random compact sets in Banach space, Math. Proc. Cambridge Phil. Soc. 97 (1985) 151– 158. [24] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409–422. [25] H. Ra˚dstro¨m, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169. [26] E.E. Robbins, On the measure of a random set, Ann. Math. Stat. 14 (1944) 70–74. [27] E.E. Robbins, On the measure of a random set II, Ann. Math. Stat. 15 (1945) 342–347. [28] E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, vol. 508, Springer-Verlag, Berlin, 1976. [29] R.L. Taylor, Stochastic convergence of weighted sums of random element in linear spaces, Lecture Notes in Mathematics, vol. 672, Springer-Verlag, Berlin, 1978. [30] R.L. Taylor, H. Inoue, Convergence of weighted random sets, Stoch. Anal. Appl. 3 (3) (1985) 379–396. [31] R.L. Taylor, H. Inoue, Law of large numbers for random sets, Random set: theory and applications, vol. 97, Springer, New York, 1997. [32] R.L. Taylor, Y. Seymour, Y. Chen, Strong laws of large numbers for fuzzy random sets, University of Georgia, Department of Statistics, Tech. Report 7, 2000. [33] R.L. Taylor, A.N. Vidyashankar, Y. Chen, Limit theorems for exchangeable random elements and exchangeable random sets, Asymp. Stat. Prob. (2000) 367–378. [34] P. Teran, I. Molchanov, The law of large numbers in a metric space with a convex combination operation, J. Theor. Prob. 19 (2006) 875–898. [35] A. Uemura, A law of large numbers for random sets, Fuzzy Sets Syst. 59 (1993) 181–188. [36] S. Utev, M. Peligrad, Maximal inequalities and an invariance principle for a class of weakly dependent random variables, J. Theor. Prob. 16 (2003) 101–115. [37] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [38] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Inform. Sci. 172 (2005) 1–40. [39] L.X. Zhang, J.W. Wen, Strong limit theorems for B-valued random fields, Chin. Ann. Math. 22A (2001) 205–216 (in Chinese).
