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The law of large numbers and renewal process for T-related weighted fuzzy numbers on Rp
Information Sciences 228 (2013) 45–60
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The law of large numbers and renewal process for T-related weighted fuzzy numbers on Rp Dug Hun Hong Department of Mathematics, Myongji University, Yongin, Kyunggido 449-728, South Korea
a r t i c l e
i n f o
Article history: Received 28 April 2011 Received in revised form 19 October 2012 Accepted 9 December 2012 Available online 19 December 2012 Keywords: Fuzzy numbers Law of large numbers Continuous Archimedean t-norm Fuzzy renewal process Possibility measure Chance measure
a b s t r a c t This paper considers the law of large numbers for T-related weighted fuzzy variables whose underlying spaces are Rp . We provide new sufficient conditions that do not depend on the shape of fuzzy numbers and the additive generator of a t-norm T for the law of large numbers for T-related weighted fuzzy variables. The results provide a far-reaching generalization of previous findings on the law of large numbers for T-related fuzzy numbers. In addition, we prove T-related fuzzy renewal theorems in which the inter-arrival time is characterized as weighted fuzzy numbers under t-norm-based fuzzy operations on Rp by using the law of large numbers for weighted fuzzy variables on Rp . We also investigate the law of large numbers and the renewal process for T-related fuzzy numbers on Rp with respect to the credibility measure, the chance measure, and the expected value. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The theory of fuzzy sets, introduced by Zadeh [32,33], has been widely examined and applied to statistics and possibility theory in recent years. Since Puri and Ralescu’s [28] introduction of the concept of fuzzy random variables, there has been growing interest in fuzzy variables. A number of studies [11–15,27,30,34–36] have investigated renewal theory in the fuzzy random environment based on the concept of fuzzy variable and fuzzy random variable. All of these studies used min-normbased fuzzy operations. In general, we can consider the extension principle realized by the means of some t-norm. Renewal theory is closely related to the law of large numbers. In 1982, Badard [1] proved the law of large numbers for fuzzy numbers with a common spread when Tðu; v Þ ¼ uv . In 1992, Fullér [5] proved the law of large numbers for a sequence of mutually T-related symmetric triangular fuzzy numbers with a common spread when Tðu; v Þ 6 H0 ðu; v Þ ¼ uv =ðu þ v uv Þ for all 0 6 u; v 6 1. A number of studies have examined various types of the law of large numbers for T-related fuzzy variables, including Williamson [31], Fullér and Triesch [6], Triesch [29], Marková [22], Hong and Lee [9], Hong and Ro [8], and Hong and Ahn [10]. Recently, Hong [11] considered a renewal reward process in which the inter-arrival time and the reward are modeled as fuzzy variables by using t-norm-based fuzzy operations and proposed a fuzzy renewal theorem and a fuzzy renewal reward theorem through the necessity measure. In [12], Hong evaluated fuzzy versions of Blackwell’s theorem by considering the necessity measure and the expected value of fuzzy variables. Wang et al. [30] obtained some limit theorems by considering the chance measure and the expected value for the sum of fuzzy random variables on the basis of continuous Archimedean tnorm-based arithmetic operations. Most of the results were based on R. In this paper, we first examine the law of large numbers for T-related weighted fuzzy numbers on Rp by using new methods. The results generalize most of the previous findings on the law of large numbers for T-related fuzzy numbers on R. Based E-mail address: [email protected] 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.12.016
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D.H. Hong / Information Sciences 228 (2013) 45–60
on these results, we examine the fuzzy renewal process for T-related fuzzy numbers on Rp with respect to the chance measure and the expected value. In Section 2, we provide the definitions of the continuous t-norm, a representation of the Archimedean t-norm, and t-norm-based fuzzy operations on a separable Banach space. In Section 3, we discuss the law of large numbers for weighted fuzzy numbers on Rp . We provide two convergence theorems for T-related weighted fuzzy numbers on Rp with respect to the general continuous t-norm and the continuous Archimedean t-norm. In Section 4, we discuss the law of large numbers for T-related fuzzy numbers with random centers and random spreads. In Section 5, we discuss the renewal process by considering the fuzzy inter-arrival time and the fuzzy reward on Rp . We derive a fuzzy renewal theorem for the rate of a renewal process having fuzzy inter-arrival times by using the law of large numbers for fuzzy numbers. In Section 6, we consider the law of large numbers and the renewal process for T-related fuzzy numbers with respect to the credibility measure. In Section 7, we consider the law of large numbers and the renewal process for T-related fuzzy numbers with respect to the chance measure and the expected value. Finally, in Section 8, some conclusion are given. 2. Preliminaries Let S be a separable Banach space with the norm k k; KðSÞ denote the class of nonempty compact convex subsets of S, and F ðSÞ denote the class of upper semicontinuous elements V of ½0; 1S such that the a-level sets V a belong to KðSÞ for all a 2 ½0; 1 with V a ¼ fx 2 SjVðxÞ P ag for a 2 ð0; 1, and V 0 ¼ clfx 2 SjVðxÞ > 0g. The space KðSÞ can be endowed with a linear structure induced by the scalar product and the Minkowski addition, that is,
kA ¼ fkaja 2 Ag; A þ B ¼ fa þ bja 2 A; b 2 Bg; for all A; B 2 KðSÞ, and k 2 R. If dH is the Hausdorff metric on KðSÞ, which for A; B 2 KðSÞ is given by
dH ðA; BÞ ¼ max sup inf ka bk; sup inf ka bk ; a2A
b2B
b2B
a2A
then ðKðSÞ; dH Þ is a complete and separable metric space ([3]). The norm of an element of KðSÞ is denoted by
kAk ¼ dH ðA; f0gÞ ¼ supfkxkjx 2 Ag: On the other hand, F ðSÞ can be endowed with the metric d1 defined ([3]) as follows:
d1 ðV; WÞ ¼ sup dH ðV a ; W a Þ;
for all V; W 2 F ðSÞ:
a2ð0;1
Here ðF ðSÞ; d1 Þ is a complete metric space ([3]). Recall that a triangular norm (or a t-norm) is a commutative monoid operation in ½0; 1 with neutral element 1 and is monotonic (non-decreasing) when viewed as a bivariate function. Now suppose that Ai 2 F ðSÞ; i ¼ 1; . . . ; n and a t-norm T ([16]) are given. Then the T-sum A1 þ þ An 2 F ðSÞ is defined via the sup-T extension principle by
ðA1 þ þ An ÞðzÞ ¼
sup TðA1 ðx1 Þ; . . . ; An ðxn ÞÞ; z 2 S: x1 þþxn ¼z
Following Fullér and Keresztfalvi [7], if T is upper semicontinuous then the equality
½A1 þ þ An a ¼
[
½A1 a1 þ þ ½An an ; a 2 ð0; 1
ð1Þ
Tða1 ;...;an ÞPa
holds. For the special case in which T ¼ min, the addition is denoted by . Then it holds that
ðkVÞa ¼ kV a ; ½A1 An a ¼ ½A1 a þ þ ½An a ;
ð2Þ
for all k 2 R, and a 2 ½0; 1 [4]. Recall that a continuous t–norm T is called Archimedean if and only if Tðx; xÞ < x for all x 2 ð0; 1Þ. A well-known theorem ([16]) asserts that for each continuous Archimedean t-norm there exists a continuous, decreasing function f : ½0; 1 ! ½0; 1 with f ð1Þ ¼ 0 such that
Tðx1 ; . . . ; xn Þ ¼ f ½1 ðf ðx1 Þ þ þ f ðxn ÞÞ for all xi 2 ð0; 1Þ; 1 6 i 6 n. Here f ½1 : ½0; 1 ! ½0; 1 is defined by
( f
½1
ðyÞ ¼
f 1 ðyÞ for y 2 ½0; f ð0Þ; 0
if y > f ð0Þ:
The function f is called the additive generator of T. Note that if a continuous t-norm T has an additive generator f, then this additive generator is uniquely determined up to a non-zero positive multiplicative constant.
D.H. Hong / Information Sciences 228 (2013) 45–60
47
Because f is continuous and decreasing, f ½1 is also continuous and non-increasing, we have from (1)
[
½A1 þ þ An a ¼
½A1 a1 þ þ ½An an :
ð3Þ
f ða1 Þþþf ðan Þ6f ðaÞ
Example 1. Examples of continuous Archimedean t-norms with additive generators are listed as follows: (a) Dombi t-norm D; p > 0:
Dp ðx; yÞ ¼
1 1
1 þ ðð1x 1Þp þ ð1y 1Þp Þp
with the additive generator f ðxÞ ¼ ðð1 xÞ=xÞp . (b) Hamacher t-norm H; p P 0:
Hp ðx; yÞ ¼
xy p þ ð1 pÞðx þ y xyÞ
with the additive generator f ðxÞ ¼ lnððp þ ð1 pÞxÞ=xÞ. (c) Sklar t-norm S; p > 0: 1
Sp ðx; yÞ ¼ ðxp þ yp 1Þ p
with the additive generator f ðxÞ ¼ ð1=pÞðxp 1Þ. (d) Frank t-norm F; p > 0; p – 1:
ðpx 1Þðpy 1Þ F p ðx; yÞ ¼ log p 1 þ p1 with the additive generator f ðxÞ ¼ log p ððp 1Þ=ðpx 1ÞÞ. From a representation theorem in topological semigroup theory [4], every continuous t-norm is the ordinal sum of a family of Archimedean t-norms. Namely, there exists a family of subintervals f½mi ; ni gi2I having non-overlapping interiors in ½0; 1 such that the following assertions hold: (i) The restriction of T to ½mi ; ni ½mi ; ni is T i Ui , where T i is an Archimedean t-norm and /i is the natural homomorphism
Ui : ðx; yÞ 2 ½mi ; ni ½mi ; ni #
x mi y mi ; ni ni ni mi
¼ ð/i ðxÞ; /i ðyÞÞ
and 0=0 ¼ 0 by convention. (ii) Elsewhere, T is the minimum. Because there are at most countably many subintervals for which ai – bi ; ½0; 1 can be decomposed into a countable union of closed subintervals with non-overlapping and non-empty interiors where either ‘T is Archimedean’ or ‘T ¼ min’ in the sense of (i). We denote KA for A 2 F ðSÞ by ½KAa ¼ ½Ani if a 2 ½mi ; ni for some i, ½KAa ¼ ½Aa , otherwise. A fuzzy variable (or a fuzzy set) on Rp is a function from Rp to ½0; 1. Definition 1 [5]. A function A 2 F ðRp Þ is called a fuzzy number (a special type of fuzzy variable) on Rp if there exists a unique element m0 2 Rp satisfying Aðm0 Þ ¼ supx AðxÞ ¼ 1. The number m0 ¼ mðAÞ is called the modal value of A. 3. The law of large numbers for weighted fuzzy numbers on Rp Throughout this section, let S ¼ Rp with the Euclidean norm k k. Let A be a fuzzy number on Rp and a 2 R1 . Then aA is a fuzzy number on Rp such that
aAðxÞ ¼ A
x ; thatis; ½aAa ¼ a½Aa ; a 2 ½0; 1: a
In addition, if B is a fuzzy number on Rp and c 2 Rp ; a > 0 such that BðxÞ ¼ aAðx cÞ, then
½Ba ¼ ½aAa þ c ¼ a½Aa þ c; a 2 ½0; 1: Let fbn g be a sequence of real numbers and fBn g be a sequence of fuzzy numbers. Then fbn Bn g is called a sequence of weighted fuzzy numbers.
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D.H. Hong / Information Sciences 228 (2013) 45–60
For a fuzzy number n and any subset D Rp , the quantity
Nesfn 2 Dg :¼ 1 supnðxÞ :¼ 1 Posfn 2 Dc g xRD
is considered to measure the necessity of n belonging to D ([29]). Note that
Nesfn 2 Dg 6 Posfn 2 Dg: Following Fuller [5], we say that a sequence of fuzzy numbers fAn g obeys the law of large numbers if for all
Sn mðSn Þ
< ¼ limn!1 NesfðSn =n 2 BðmðSn Þ=n; Þg ¼ 1 limn!1 Nes
n
>0
where Bða; Þ ¼ fx 2 Sjka xk < g and Sn ¼ A1 þ A2 þ þ An . We note that if n be a fuzzy number with mðnÞ ¼ 0, then
Nesfknk < g ¼ Nesfn 2 Bð0; Þg > 1 a () Posfn 2 Bð0; Þc g < a () dH ðf0g; ½na Þ ¼ k½na k < : The following result is immediate from this fact. Proposition 1. A sequence of fuzzy numbers fAn g obeys the law of large numbers if and only if for any 0 < a < 1
limn!1 k½ðSn mðSn ÞÞ=na k ¼ 0 The following result is known. Proposition 2 [3]. Let A 2 F ðRp Þ and write C a ¼ ½Aa for a 2 ½0; 1. Then
dH ðC b ; C a Þ ! 0asb " a: We first introduce the following sufficient condition of the law of large numbers for fuzzy variables by using Proposition 1. Theorem 1. Let A be a fuzzy number on Rp with mðAÞ ¼ 0; An be a fuzzy number such that An ðxÞ ¼ an Aðx cn Þ, where cn 2 Rp ; an > 0; n ¼ 1; 2; . . ., denotes a sequence of fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm with P the additive generator f is given, limsupð ni¼1 ai =nÞ 6 M, for some M > 0, and limðmax16i6n ai =nÞ ¼ 0. Then we have, for all > 0
Sn mðSn Þ
< ¼1 limn!1 Nes
n
P Proof. Let 0 < a < 1; > 0 be given and f ð1 Þ ¼ t 0 . Suppose that ni¼1 f ðai Þ 6 f ðaÞ and let Ha; ¼ fkjf ðak Þ > t 0 g. Then the number of Ha; is less than or equal to N where N is the smallest natural number bigger than f ðaÞ=t 0 . We note that if k R Ha; , then ak P 1 . Then, we have from (3)
½a1 A þ þ an Aa ¼
[ Pn i¼1
ð½a1 Aa1 þ þ ½an Aan Þ
f ðai Þ6f ðaÞ
½a1 A1 þ þ ½an A1 þ
[ Pn i¼1
f ðxi Þ6f ðaÞ
¼ a1 ½A1 þ þ an ½A1 þ NBð0; tn Þ ¼
[ Pn 0 @
i¼1
f ðai Þ6f ðaÞ
X
0 @
X
½ak A1 þ
kRHa;
X
1 ½ak A0 A
k2Ha;
1
½ak A0 A ½a1 A1 þ þ ½an A1 þ NBð0; tn Þ
k2Ha;
X
ai ½A1 þ NBð0; t n Þ;
16i6n
where t n ¼ max16i6n ai k½A0 k and the second equality above comes from the convexity of ½A0 . Hence, we have, for large n,
Pn
Sn mðSn Þ 1
¼ ½a1 A þ þ an A 6 i¼1 ai k½A k þ N B 0; t n 6 Mk½A k þ N t n : a
1 1
n n n n n a
The first term goes to 0 because is arbitrary and lim!0 k½A1 k ¼ 0 by Proposition 2 and the second term goes to 0 as n ! 1 by assumption. This completes the proof. P In Theorem 1, if fan g is bounded, that is, an 6 K; n 2 N, for some K > 0, then two conditions limsupð ni¼1 ai =nÞ 6 M, for some M > 0, and limðmax16i6n ai =nÞ ¼ 0 are satisfied. Thus, the following result is immediate. h
D.H. Hong / Information Sciences 228 (2013) 45–60
49
Corollary 1. Let A be a fuzzy variable on Rp with mðAÞ ¼ 0 and let An be a fuzzy variable such that An ðxÞ ¼ an Aðx cn Þ, where cn 2 Rp ; an > 0; n ¼ 1; 2; . . ., denotes a sequence of fuzzy variables on Rp . Suppose that a continuous Archimedean t-norm with additive generator f is given and that fan g is bounded, that is, an 6 K; n 2 N, for some K > 0. Then the sequence fAn g obeys the law of large numbers. Note 1. One of the most generalized result of a law of large numbers for fuzzy numbers of Fullér [6] was proved by Marková [22, Theorem 2]. This is the case of fuzzy variables on R1 with cn ¼ 0; an 6 K for all n ¼ 1; 2; . . ., in Corollary 1. We now consider a necessary condition of the law of large numbers for weighted fuzzy numbers.
Theorem 2. Let A be a fuzzy number on Rp with mðAÞ ¼ 0 and let An be a fuzzy number such that An ðxÞ ¼ an Aðx cn Þ, where cn 2 Rp ; an > 0; n ¼ 1; 2; . . ., denotes a sequence of fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm with an additive generator f is given and that the sequence fAn g obeys the law of large numbers. Then we have
limðmax16i6n ai =nÞ ¼ 0: Proof. We have from (3)
½a1 A þ þ an Aa ¼
[ Pn i¼1
ð½a1 Aa1 þ þ ½an Aan Þ ¼
f ðai Þ6f ðaÞ
[ Pn i¼1
ða1 ½Aa1 þ þ an ½Aan Þ:
f ðai Þ6f ðaÞ
Let max16i6n ai ¼ an and choose an ¼ a, and ai ¼ 1; i – n . Then we have
Pn
a ¼ f ðaÞ and
i¼1 f ð i Þ
½a1 A þ þ an Aa an ½Aan ¼ max16i6n ai ½Aa ; and hence
Sn mðSn Þ 1
¼ ½a1 A þ þ an A P max16i6n ai k½A k: a
a
n n n a Therefore the result follows from the assumption. h Combining Theorems 1 and 2, we have the following result. Theorem 3. Let A be a fuzzy number on Rp with mðAÞ ¼ 0 and let An be a fuzzy number such that An ðxÞ ¼ an Aðx cn Þ, where cn 2 Rp ; an > 0; n ¼ 1; 2; . . ., denotes a sequence of fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm with an P additive generator f is given and that limsupð ni¼1 ai =nÞ 6 M, for some M > 0. Then the sequence of fuzzy numbers fAn g obeys the law of large numbers if and only if limðmax16i6n ai =nÞ ¼ 0.
Note 2. This is a different type of equivalent condition from Hong and Ahn [10, Theorem 3] for obeying the law of large numbers for fuzzy numbers. The conditions have nothing to do with the shape of fuzzy numbers and the additive generator of the continuous Archimedean t-norm. The following result shows the convergence of sequences of fuzzy variables with respect to a general continuous t-norm. From this result, we can see that the Archimedean property is essential for the law of large numbers. We need the following lemma which is easy to prove.
Lemma 1. Let Ai 2 Rp ; i ¼ 1; 2; 3 such that A1 A2 A3 . Then we have
dH ðA1 ; A2 Þ 6 dH ðA1 ; A3 Þ: Theorem 4. Let A be a fuzzy variable on Rp with mðAÞ ¼ 0; An ¼ an A; n ¼ 1; 2; . . ., where A 2 F ðRp Þ; an > 0, and T be a continuous P t-norm. Suppose that limð ni¼1 ai =nÞ ¼ 1 and limðmax16i6n ai =nÞ ¼ 0. Then we have for all 0 < a 6 1,
lim dH
n!1
Sn ; ½KAa ¼ 0: n a
Proof. Let T be a continuous t-norm such that the restriction of T to ½mi ; ni ½mi ; ni is indicated by T i /i , where T i is an Archimedean t-norm with an additive generator fi . If a R [ðmi ; ni Þ, because T ¼ min except on [fðmi ; ni Þ ðmi ; ni Þg, then we clearly have
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D.H. Hong / Information Sciences 228 (2013) 45–60
[
½a1 A þ þ an Aa ¼
! n X ai ½Aa ;
ð½a1 Aa1 þ þ ½an Aan Þ ¼ ½a1 Aa þ þ ½an Aa ¼
Tða1 ;...;an ÞPa
i¼1
and hence the result follows immediately. Now, let mj < a < nj for some j. Note that if Tða1 ; . . . ; an Þ P a, then there exists ða01 ; . . . ; a0n Þ such that ai P a0i ; mj 6 a0i 6 nj ; i ¼ 1; . . . ; n and Tða01 ; . . . ; a0n Þ P a. Then, because ½ai Aai ½ai Aa0 ; i ¼ 1; . . . ; n, we have i
[
½a1 A þ þ an Aa ¼
ð½a1 Aa1 þ þ ½an Aan Þ
Tða1 ;...;an ÞPa
¼
[
Pn
[ T j ða01 ;...;a0n ÞPa
ð½a1 Aa0 þ þ ½an Aa0n Þ 1
ð½a1 Aa0 þ þ ½an Aa0n Þ
ð4Þ
1
l¼1
fj ð/j ða0 ÞÞ6fj ð/j ðaÞÞ l
Let > 0 be a number such that nj > mj and let fj ð/j ðnj ÞÞ ¼ t0 ; Ha; ¼ fkjfj ð/j ða0k ÞÞ > t0 g. Then the number of Ha; is less than N where N is the smallest natural number bigger than fj ð/j ðaÞÞ=t 0 . Note that if k R Ha; , then a0k P nj . Then, we have from (4) that
[
½a1 A þ þ an Aa ¼
ð½a1 Aa1 þ þ ½an Aan Þ
Tða1 ;...;an ÞPa
[ Pn
0 @
f ð/ ða0 ÞÞ6fj ð/j ðaÞÞ l¼1 j j l
[
ð½a1 Aa0 þ þ ½an Aa0n Þ
Pn
1
f ð/ ða0 ÞÞ6fj ð/j ðaÞÞ l¼1 j j l
X
½ak Anj þ
kRHa;
½a1 Anj þ þ ½an Anj þ
X
1
½ak A0 A
k2Ha;
[ Pn l¼1
0 @
fj ð/j ða0l ÞÞ6fj ð/j ðaÞÞ
X
1 ½ak A0 A
k2Ha;
½a1 Anj þ þ ½an Anj þ NBð0; t n Þ ¼ a1 ½Anj þ þ an ½Anj þ NBð0; t n Þ
n X ak ½Anj þ NBð0; tn Þ;
ð5Þ
k¼1
where t n ¼ max16k6n ak k½A0 k. On the other hand,
½a1 A þ þ an Aa ½a1 A þ þ an Anj ¼ ½a1 Anj þ þ ½an Anj ¼
n X
ak ½Anj :
ð6Þ
k¼1
Then
dH
Pn Pn Pn Pn Sn k¼1 ak k¼1 ak k¼1 ak k¼1 ak ; ½KAa ¼ dH ½Aa ; ½Anj 6 dH ½Anj ; ½Anj þ dH ½Anj ; ½Aa n a n n n n Pn Pn Pn a a a t n k k k k¼1 k¼1 k¼1 ½Anj ; ½Anj þ dH ½Anj ; ½Anj þ NB 0; 6 dH n n n n Pn Pn Pn a a a t n k¼1 k k¼1 k k¼1 k ½Anj ; ½Anj þ dH ½Anj ; ½Anj þ N 6 dH n n n n Pn Pn ak tn k¼1 ak dH ð½Anj ; ½Anj Þ þ N ; 6 k¼1 1 k½A0 k þ n n n
where the second inequality comes from (5), (6) and Lemma 1 and the third inequality comes from Proposition 2.4.1 [4]. The first and third terms go to 0 as n ! 1 by assumption, and the second term goes to 0 since is arbitrary and lim!0 dH ð½Ani ; ½Ani Þ ¼ 0 by Proposition 2, which completes the proof. h Note 3. The case of fuzzy variables on R1 with an ¼ 1 for all n ¼ 1; 2; . . ., which was proved by Marková ([22] Theorem 5) is a special case of Theorem 4.
4. The law of large numbers with random spreads and random centers e : X ! F ðRp Þ is a fuzzy random variable [16] if every a-cut of Let ðX; T ; PÞ be a probability space. The mapping Y p e e e Y ; Y a ðxÞ ¼ fx 2 R : ð Y ðxÞÞðxÞ P ag is a compact random set. For general discussions on fuzzy random variables, see Kwakernaak [17], Puri and Ralescu [28]. Let A 2 F ðRp Þ with ~ be a positive real valued random variable, ~c be Rp -valued random variable on ðX; T ; PÞ, and g : X ! F ðRp Þ be a mðAÞ ¼ 0; a ~ðxÞAðx ~cðxÞÞ. Then g is clearly a fuzzy random variable. function defined by gðxÞðxÞ ¼ a
D.H. Hong / Information Sciences 228 (2013) 45–60
51
Let A be a fuzzy number on Rp with mðAÞ ¼ 0. If B is a fuzzy variable on Rp and c ¼ ðc1 ; c2 ; . . . ; cp Þ 2 Rp ; a > 0 such that BðxÞ ¼ aAðx cÞ, then c is called the center of B and a is called the spread of B. In this section, we consider laws of large numbers for fuzzy numbers on Rp with random centers and spreads. The following important lemma is well known in probability theory. Lemma 2. Let fX n g1 and identically distributed (i.i.d.) random variables. Then EjX 1 j < 1 iff n¼1 be independent P n1 max16k6n jX k j ! 0 a.s. iff lim n1 nk¼1 X k ¼ E½X 1 a.s. By Lemma 2, the following is immediate from Theorem 1. 1 p ~i ðxÞg1 ~ Theorem 5. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued i.i.d. random variables, A be a fuzzy number on Rp with mðAÞ ¼ 0, and fAn g be a sequence of fuzzy random variables such that ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given. Then Ea ~1 < 1 if and only if the sequence An ðxÞðxÞ ¼ a of fuzzy numbers fAn g obeys the law of large numbers a.s., that is, for all > 0
Pn
Sn ðxÞ
~ i¼1 c i ðxÞ
< ¼ 1 a:s: limn!1 Nes
n
n We also obtain the following result if we follow the idea behind the proof in Theorem 6 [10] by using Theorem 5 and Lemma 2. 1 p ~i ðxÞg1 ~ Theorem 6. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued i.i.d. p random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fAn g be a sequence of fuzzy random variables such that ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given. Then Ea ~1 < 1 and Ejj~c1 jj < 1 if and only An ðxÞðxÞ ¼ a if for all > 0,
Sn ðxÞ
< ¼ 1 a:s: ~ limn!1 Nes
E c 1
n
~i ðxÞg1 Example 2. Let A be a fuzzy number on R2 with Aðx; yÞ ¼ maxf0; 1 ðx2 þ y2 Þg; fa i¼1 be a sequence of positive real val2 ued i.i.d. random variables which are uniformly distributed on ð0; 100, and f~ci ðxÞg1 i¼1 be a sequence of R -valued i.i.d. ran~1 < 1. Let fAn g be a sequence of fuzzy dom variables that are uniformly distributed on ½1; 3 ½2; 8. Then E~c1 ¼ ð2; 5Þ and Ea ~n ðxÞAðx ~cn ðxÞÞ. Then by Theorem 6, for any > 0, we have random variables such that An ðxÞðxÞ ¼ a
Sn ðxÞ
< ¼ 1 a:s: ð2; 5Þ limn!1 Nes
n 5. Renewal process with fuzzy inter-arrival times on Rp In this section, we assume that S ¼ Rp and consider the renewal process with fuzzy inter-arrival times. Let nn denote pdimensional times between the ðn 1Þth and nth events which are known as pdimensional inter-arrival times, n ¼ 1; 2; . . .. Define
S0 ¼ 0; Sn ¼ n1 þ n2 þ þ nn ; n P 1: If pdimensional inter-arrival times nn ; n ¼ 1; 2; . . . are fuzzy variables on Rp , then the process fnn ; n P 1g is called a pdimensional fuzzy renewal process. We assume that ½ni 0 ðRþ Þp , that is, ni is positive, i ¼ 1; 2; . . .. Let NðtÞ denote the total number of events that occurred by time t. Then we have
NðtÞ ¼ maxnP0 fnj0 < kSn k 6 tg; where kSn kðzÞ ¼ supkxk¼z Sn ðxÞ. It is clear that NðtÞ is a nonnegative integer valued fuzzy variable, and
PosfNðtÞ ¼ ng ¼ PosfkSn k 6 t < kSnþ1 kg; ( PosfNðtÞ < ng ¼ Pos
n1 [
ð7Þ
) fNðtÞ ¼ ig
¼ PosfkSn k > tg;
ð8Þ
¼ PosfkSn k 6 tg;
ð9Þ
i¼0
PosfNðtÞ P ng ¼ Pos
( 1 [
) fNðtÞ ¼ ig
i¼n
for each integer n. We call NðtÞ the fuzzy renewal variable. The following lemma is easy to check.
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D.H. Hong / Information Sciences 228 (2013) 45–60
Lemma 3. Let fni g be a sequence of fuzzy variables on Rp obeying the law of large numbers. If mðSn Þ=n ! c as n ! 1, then for any > 0,
Sn
< ¼ 1: limn!1 Nes
c
n Theorem 7 (Fuzzy Renewal Theorem in Necessity). Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ ai Aðx ci Þ; i ¼ 1; 2; . . . be a sequence of positive fuzzy variables on Rp , and NðtÞ be a fuzzy renewal variable. Suppose that a continuous Archimedean tP norm is given, that limsupð ni¼1 ai =nÞ 6 K, for some K 2 R, and limðmax16i6n ai =nÞ ¼ 0. If mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1, then for any > 0,
NðtÞ 1 < ¼1 limt!1 Nes t kck Proof. Because ni ; i ¼ 1; 2; . . ., obeys the law of large numbers by Theorem 1, and mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1, we have by Lemma 3, for any > 0,
Sn
< ¼ 1: c limn!1 Nes
n
Noting that
Sn
jkSn k nkckj
< ; Nes
n c < 6 Nes n we have
limn!1 Nes
jkSn k nkckj < ¼ 1: n
Note that it suffices to prove that for small
ð10Þ
> 0,
NðtÞ 1 < limt!1 Nes ¼1 t kck Because NðtÞ is a fuzzy variable taking an integer value, we have
NðtÞ 1 NðtÞ 1 NðtÞ 1 < P 1 Pos _ Pos Nes < P þ t kck t kck 2 t kck 8 9 < = ik > t ¼ 1 Pos kSh 1 t : ; kck 2 8 9 < = i k 6 t ðwhere ½ is the smallest integer greater than or equal to Þ _ Pos kSh 1 þ t : ; kck 8 h 9 8 h 9 ik ik kS kS > > > > 1 t 1 þ t < = < = kck 2 kck t t i > h i _ Pos h i 6 h i ¼ 1 Pos h > > 1 1 1 1 : kck ; : kck ; 2 t 2 t > þ t þ t > kck kck where the first equality comes from (8) and (9). Then, because
limt!1 h
1 kck
t
1 i¼ 1 > kck; ð 2Þ t kck 2
and there exists an
limt!1 h
0 > 0 such that
1 > kck þ 0 1 ðkck 2Þ we have, for large t,
and
1 < kck 0 ; 1 ðkck þ Þ
t
1 kck
1 i¼ 1 < kck; ð þ Þ þ t kck
D.H. Hong / Information Sciences 228 (2013) 45–60
53
8 h 9 8 h 9 ik ik kS kS > > > > 1 t 1 þ t < = < = NðtÞ kck 2 kck 1 0 0 h i h i < > kck þ < kck P 1 Pos Nes _ Pos > > > > 1 t kck : ; : 1 þ t ; t kck
2
kck
kSn k kSm k R ½kck 0 ; kck þ 0 _ Pos R ½kck 0 ; kck þ 0 : P 1 Pos n m 1 1 þ tas mÞ tas n and ½ ðregarding½ kck 2 kck
Because
kSn k jkSn k nkckj R ½kck 0 ; kck þ 0 ¼ 1 Nes < 0 ; Pos n n goes to 0 as t ! 1 by (10), and similarly, PosfkSm k=m R ½kck 0 ; kak þ 0 g goes to 0 as t ! 1. Hence the proof is complete. h The following result is immediate from Theorem 7. Corollary 2. Let A be a fuzzy number on Rp with mðAÞ ¼ 0 and ni ðxÞ ¼ Aðx ci Þ; i ¼ 1; 2; . . . be a sequence of positive fuzzy variables on Rp . Suppose that a continuous Archimedean t-norm is given and that mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then we have, for any > 0,
NðtÞ 1 < ¼ 1: limt!1 Nes t kck Theorem 8. Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ ai Aðx ci Þ; i ¼ 1; 2; . . . be a sequence positive fuzzy variables on Rp obeying the law of large numbers, and NðtÞ be a fuzzy renewal variable. If mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1, then for any > 0,
SNðtÞ
< ¼ 1: limt!1 Nes
c
NðtÞ
Proof. Because it suffices to prove that for small
>0
SNðtÞ
limt!1 Nes
NðtÞ c < ¼ 1; we assume that 0 < < 1=kck. Then we have
SNðtÞ
< ¼ 1 Pos SNðtÞ c P ¼ 1 supn Pos SNðtÞ c P ; NðtÞ ¼ n Nes
c
NðtÞ
NðtÞ
NðtÞ
i Pos Sn c P ; NðtÞ ¼ n ¼ 1 sup h
n
1 t; 1 þ t n2 kck
kck
i Pos Sn c P ; NðtÞ ¼ n
1 t; 1 þ t n nR kck kck
i Pos Sn c P _ sup h i PosfNðtÞ ¼ ng P 1 sup h
1 t; 1 þ t 1 t; 1 þ t n n2 nR kck kck kck kck _ sup h
¼ 1 At _ Bt ; where
NðtÞ 1 1 1 < ¼ 1 Nes ! 0ast ! 1ðby Theorem7Þ t; þ t Bt ¼ Pos NðtÞ R kck kck t kck and
At ¼ sup h n2
1 kck
t;
i Pos Sn c P ¼ 1 inf h i Nes Sn c < ! 0ast
1 þ t 1 t; 1 þ t n n n2
kck
! 1ðby Lemma3Þ which completes the proof. h
kck
kck
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D.H. Hong / Information Sciences 228 (2013) 45–60
6. Results from the credibility measure In this section, we consider the law of large numbers and the renewal process for T-related fuzzy numbers with respect to the credibility measure. The credibility of n belonging to D is defined as
Crfn 2 Dg ¼ ð1=2ÞðPosfn 2 Dg þ Necfn 2 DgÞ: Before we present the expected value of a fuzzy variable, we first recall the definition of Choquet integral. This kind of integral was first introduced in [2], and later was restudied in the field of fuzzy measure theory by some researchers such as Murofushi and Sugeno [23,24], Murofushi, Sugeno and Machida [25], and Narukawa, Murofushi and Sugeno [26]. Let ðX; B; lÞ be a fuzzy measure space. The Choquet integral of a nonnegative measurable function with respect to a fuzzy measure l is defined as
ðCÞ
Z
fdl ¼
Z
1
lðfxjf ðxÞ P rgÞdr 0
If
l is a finite fuzzy measure, then the Choquet integral of a measurable function with respect to l is defined as ðCÞ
Z
fdl ¼ ðCÞ
Z
f þ dl ðCÞ
Z
f dld
where f þ ¼ f _ 0; f ¼ ðf ^ 0Þ, and ld is the dual of l in the sense that lðAÞ ¼ lðXÞ ld ðAC Þ. And from the measure-theoretic interpretation of Choquet integral, it is usually regarded as the generalization of usual mathematical expectation. Therefore, motivated by the idea of Choquet integral, E½n(see [21]) is defined as
E½n ¼
Z
1
Crðn P rÞdr
0
Z
0
Crðn 6 rÞdr 1
provided that at least one of the two integrals is finite. In particular, if n is a nonnegative fuzzy variable (i.e., Crfn < 0g ¼ 0), R1 then E½n ¼ 0 Crfn P rgdr. Definition 2. The sequence fnn g converges in necessity to n if for any
> 0,
limn!1 Necfknn nk < g ¼ 1: Definition 3. The sequence fnn g converges in credibility to n if for any
> 0,
limn!1 Crfknn nk P g ¼ 0: Note that the condition
lim Nesfnn 2 Dn g ¼ 1
n!1
implies
Crfnn 2 Dcn g 6 Posfnn 2 Dcn g ¼ 1 Nesfnn 2 Dn g ! 0: Hence we have the following result. Proposition 3. If the sequence fnn g converges in necessity to n, then fnn g converges in credibility to n. By Lemma 3 and Proposition 3, the following result is immediate from Theorem 1. Theorem 9. Let A be a fuzzy number on Rp with mðAÞ ¼ 0, and nn ðxÞ ¼ an Aðx cn Þ; cn 2 Rp ; an > 0; n ¼ 1; 2; . . . denote a sequence P of fuzzy variables on Rp . Suppose that a continuous Archimedean t-norm is given, limsupð ni¼1 an =nÞ 6 K, for some K 2 R; limðmax16i6n ai =nÞ ¼ 0, and mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then we have, for any > 0,
Sn
P ¼ 0: limn!1 Cr
c
n
By Proposition 3, the following is immediate from Theorem 7. Theorem 10. Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ ai Aðx ci Þ; n ¼ 1; 2; . . . be a sequence of positive fuzzy variP ables on Rp , and NðtÞ be fuzzy renewal variable. Suppose that a continuous Archimedean t-norm is given, limsupð ni¼1 an =nÞ 6 K, for some K 2 R; limðmax16i6n ai =nÞ ¼ 0, and limn!1 ðc1 þ þ cn Þ=n ¼ c. Then
D.H. Hong / Information Sciences 228 (2013) 45–60
55
NðtÞ 1 P ¼ 0: limn!1 Cr t kck The following result is immediate from Theorem 10. Corollary 3. Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ Aðx ci Þ; n ¼ 1; 2; . . . be a sequence of positive fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm is given and that mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then fNðtÞ=tg con1 verges in credibility to kck . Lemma 4. Let A be a fuzzy number on Rp with mðAÞ ¼ 0, and ni ðxÞ ¼ ai Aðx ci Þ; i ¼ 1; 2; . . . be a sequence of positive fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm is given, ða1 þ þ an Þ=n ! a as n ! 1, and mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then for large r > 0, and for large t > 0, we have
Cr
NðtÞ P r ¼ 0: t
Proof. Because ni ðxÞ ¼ ai Aðx ci Þ; i ¼ 1; 2; . . . are positive fuzzy variables, limn!1 ða1 þ þ an Þ=n ¼ a, and limn!1 ðc1 þ þ cn Þ=n ¼ c, there exists 1 ; 2 > 0 such that 0 < 2 ¼ inf fjjxjj : x 2 ða þ 1 Þ½A0 þ cg. Let I be an indicator function. We consider that
S½rt
NðtÞ
6 t=½rt 6 Pos S½rt 6 t P r ¼ PosfNðtÞ P rtg ¼ PosfNðtÞ P ½rtg ¼ PosfkS½rt k 6 tg ¼ Pos
Pos
t tr ½rt ½rt 8 ½rt 9 X > > > > > > Iai ½A0 þci ðzÞ > >
<
S½rt 1 S½rt 1 1= i¼1
¼ Pos 6 ¼ sup 6 sup ðzÞ : kzk 6 : kzk 6 > r r r> ½rt ½rt ½rt > > > > > > : ; where ½rt is the smallest integer such that ½rt P rt. Because, for large t > 0
P½rt
i¼1 I ai ½A0 þci ðzÞ
½rt
¼ IP½rt a
P½rt ðzÞ 6 Iðaþ1 Þ½A0 þc ðzÞ c
i¼1 i ½A þ 0 ½rt
i¼1 i ½rt
and
2 sup Iðaþ1 Þ½A0 þc ðzÞ : kzk 6 ¼ 0; 2 we have, for large t > 0 and for r > 2=2 ,
NðtÞ P r ¼ 0; Pos t which proves the result noting that
Cr
NðtÞ NðtÞ P r 6 Pos Pr : t t
h Theorem 11 (Fuzzy Renewal Theorem in Credibility). Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ ai Aðx ci Þ, be a sequence of positive fuzzy variables on Rp , and NðtÞ be fuzzy renewal variable. Suppose that a continuous Archimedean t-norm given, limðmax16i6n ai =nÞ ¼ 0; ða1 þ þ an Þ=n ! a as n ! 1, and mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then
limt!1
E½NðtÞ 1 ¼ : t kck
Proof. According to Theorems 10, we know that fNðtÞ=tg converges in credibility to 1=kck. Then, it follows from Liu [20, Theorem 3.54] that for any r 2 R,
limt!1 CrfNðtÞ=t P rg ¼ Cr
1 Pr : kck
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D.H. Hong / Information Sciences 228 (2013) 45–60
By Lemma 4, we note that, for large t > 0 and for r > 2=,
Cr
NðtÞ NðtÞ P r 6 Pos P r ¼ 0: t t
As a consequence, by Lebesgue dominated convergence theorem, we have
limt!1
E½NðtÞ ¼ limt!1 t
Z
1
CrfNðtÞ=t P rgdr ¼
0
1 : kck
By Proposition 3 and Lemma 2, the following is immediate from Theorem 7. h 1 p ~i ðxÞg1 ~ Theorem 12. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given. If Ea ~1 < 1, and Ejj~c1 jj < 1, then that nn ðxÞðxÞ ¼ a
NðxÞðtÞ 1 limt!1 Cr t jjE~c
1
P ¼ 0 a:s: jj
By Lemma 2, the following is immediate from Theorem 11. ~i ðxÞg1 Theorem 13 (Fuzzy Random Renewal Theorem in Credibility). Let fa i¼1 be a sequence of positive real valued i.i.d. random p variables, f~ci ðxÞg1 be a sequence of R -valued i.i.d. random variables, A be a fuzzy number on Rp with mðAÞ ¼ 0, and fnn g be a i¼1 ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean tsequence of positive fuzzy random variables such that nn ðxÞðxÞ ¼ a ~1 < 1, and Ejj~c1 jj < 1. Then we have norm is given, Ea
limt!1
E½NðxÞðtÞ 1 ¼ a:s: t jjE~c1 jj
By Proposition 3, the following is immediate from Theorem 8. Theorem 14. Let A be a fuzzy number on Rp with mðAÞ ¼ 0, and ni ðxÞ ¼ ai Aðx ci Þ, be a sequence of positive fuzzy numbers on Rp . P Suppose that a continuous Archimedean t-norm is given, limsupð ni¼1 an =nÞ 6 K, for some K 2 R; limðmax16i6n ai =nÞ ¼ 0, and mn ¼ ðc1 þ þ cn Þ=n ! casn ! 1. Then for any > 0,
SNðtÞ
P ¼ 0: limt!1 Cr
c
NðtÞ
By Proposition 3 and Lemma 2, the following is immediate from Theorem 8. 1 p ~i ðxÞg1 ~ Theorem 15. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm f is given, Ea ~1 < 1, and Ejj~c1 jj < 1. Then for that nn ðxÞðxÞ ¼ a any > 0,
SNðxÞðtÞ
P ¼ 0 a:s: ~ limt!1 Cr
E c 1
NðxÞðtÞ
Example 3. Let A be a fuzzy number on R2 defined by Aðx; yÞ ¼ maxf0; 1 ðx2 þ y2 Þg. Then ½A0 ¼ fðx; yÞ : x2 þ y2 6 1g. Let ~i ðxÞg1 fa i¼1 be a sequence of positive real valued i.i.d. random variables that are uniformly distributed on ½1; 3 and 2 f~ci ðxÞg1 i¼1 be a sequence of R -valued i.i.d. random variables that are uniformly distributed on ½3; 7 ½3; 11. Then ~1 < 1. Let fnn g be a sequence of fuzzy random variables such that nn ðxÞðxÞ ¼ a ~n ðxÞAðx ~cn ðxÞÞ. Then E~c1 ¼ ð5; 7Þ and Ea fnn g are positive fuzzy random variables and by Theorem 6, 12 and 13, we have, for any > 0,
Sn ðxÞ
< ¼ 1 a:s:; limn!1 Nes
ð5; 7Þ
n
NðxÞðtÞ 1 limn!1 Cr pffiffiffiffiffiffi P ¼ 0 a:s: t 74 and
limt!1
E½NðxÞðtÞ 1 ¼ pffiffiffiffiffiffi a:s: t 74
D.H. Hong / Information Sciences 228 (2013) 45–60
57
7. Results from the chance measure In this Section, we consider the law of large numbers, the renewal process for T-related fuzzy numbers with respect to the chance measure. Roughly speaking, a fuzzy random variable is a measurable function from a probability space to a collection of fuzzy numbers. A fuzzy random variable is a mapping n : X ! F ðRp Þ such that for any closed subset of C of Rp ,
n ðCÞðxÞ ¼ PosfnðxÞ 2 Cg ¼ suplnðxÞ ðxÞ x2C
is a measurable function of x, where lnðxÞ is the possibility distribution function of fuzzy variable nðxÞ. A scalar expected value is often required as a surrogate for a fuzzy random variable so that a decision-maker may employ the value to make a decision. For any fuzzy random variable n; nðxÞ is a fuzzy variable for all x 2 X. And E½nðxÞ, the expected value of nðxÞ, is a random variable defined on X. If we take the value E½nðxÞ as a surrogate for the n, then we have a novel definition of the expected value as follow. Definition 4 [18]. Let n be a fuzzy random variable. The expected value of n is defined as the expected value of E½nðxÞ, i.e.,
E½n ¼
Z Z X
1
Cr ðnðxÞ P r Þdr
Z
0
0
Cr ðnðxÞ 6 rÞdr dPðxÞ
ð11Þ
1
provided that at least one of the two integrals is finite. In particular, if n is a positive fuzzy random variable, then R R1 E½n ¼ X 0 Cr ðnðxÞ P r ÞdrdPðxÞ. Definition 5 [19]. Let n be a fuzzy random variable defined on the probability space ðX; A; PrÞ. The chance of an event n 2 B is defined as
Chfn 2 Bg ¼
Z
CrfnðxÞ 2 BgdPðxÞ:
X
It has been proved that (11) is equivalent to the following form (see [19,20]):
E½n ¼
Z 0
1
Chfn P rgdr
Z
0
Chfn 6 rgdr:
1
Definition 6 [30]. The sequence fnn g converges in chance to n if for any
> 0,
limn!1 Chfknn nk P g ¼ 0: A fuzzy random variable n is positive if and only if CrfnðxÞ 6 0g ¼ 0, for any x 2 X. By Lemma 2 and the bounded convergence theorem, the following is immediate from Theorem 9.
1 p ~i ðxÞg1 ~ Theorem 16. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm f is given, Ea ~1 < 1, and Ejj~c1 jj < 1. Then for that nn ðxÞðxÞ ¼ a any > 0,
Sn
~ P ¼ 0: limn!1 Ch
E c 1
n
By the bounded convergence theorem, the following is immediate from Theorem 15. 1 p ~i ðxÞg1 ~ Theorem 17. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm f is given, Ea ~1 < 1, and Ejj~c1 jj < 1. Then for that nn ðxÞðxÞ ¼ a any > 0,
SNðtÞ
~ P ¼ 0: limt!1 Ch
E c 1
NðtÞ 1 p ~i ðxÞg1 ~ Theorem 18. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given. If Ea ~1 < 1 and Ejj~c1 jj < 1, then that nn ðxÞðxÞ ¼ a
58
D.H. Hong / Information Sciences 228 (2013) 45–60
NðtÞ 1 P ¼ 0: limt!1 Ch t jjE~c1 jj Proof. According to Theorem 12, we know that fNðxÞðtÞ=tg converges in credibility to 1=jjE~c1 jj a.s. As a consequence, by bounded convergence theorem, we have
Z NðtÞ NðxÞðtÞ 1 1 P P ¼ lim dPðxÞ ¼ 0: limt!1 Ch Cr t!1 t Ejj~c1 jj t jjE~c1 jj X h ~i ðxÞg1 Lemma 5. Let A be a fuzzy variable on Rp with mðAÞ ¼ 0; fa i¼1 be a sequence of positive real valued i.i.d. random variables, 1 p ~ fci ðxÞgi¼1 be a sequence of R -valued i.i.d. random variables, and fnn g be a sequence of positive fuzzy random variables such that ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given, 0 6 a1 ðxÞ 6 H a:s., and nn ðxÞðxÞ ¼ a jj~c1 ðxÞjj P M > 0 a:s. If M Hjj½A0 jj > 0, then for big r > 0,
NðtÞ P r ¼ 0: Ch t
Proof. Let I be an indicator function. As we have proved in Lemma 4,
Cr
NðxÞðtÞ Pr t
6 Pos
S½rt ðxÞ NðxÞðtÞ 1 P r ¼ sup ðzÞ : kzk 6 t r ½rt (P½rt
6 sup
i¼1 Iai ðxÞ½A0 þci ðxÞ ðzÞ
½rt
: kzk 6
1 r
)
8 9 < 1= P½rt P½rt 6 sup I ¼ supI ðzÞ : kzk 6 ðzÞ c ðxÞ : H½A þ i¼1 ci ðxÞ r; i¼1 i H½A þ \Bð0;1=rÞ 0
0
½rt
½rt
where ½rt is the smallest integer such that ½rt P rt. Then we have
inf fjjxjj : x 2 H½A0 þ
n X ci ðxÞ=ng P M Hjj½A0 jj > 0 a:s:; i¼1
for all n ¼ 1; 2; . . . and hence, for large r > 0, ½rt X c i ð xÞ
H½A0 þ
i¼1
½rt
\ Bð0; 1=rÞ ¼ ; a:s:
that is,
Cr
NðxÞðtÞ P r ¼ 0 a:s: t
Then we have, for large r > 0,
Z NðtÞ NðxÞðtÞ P r ¼ Cr P r dPðxÞ ¼ 0: Ch t t h ~i ðxÞg1 Theorem 19 (Fuzzy Renewal Theorem in Chance). Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, 1 p ~ fci ðxÞgi¼1 be a sequence of R -valued i.i.d. random variables, A be a fuzzy number on Rp with mðAÞ ¼ 0, and fnn g be a sequence ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is of positive fuzzy random variables such that nn ðxÞðxÞ ¼ a given, 0 6 a1 ðxÞ 6 Ha:s., and jj~c1 ðxÞjj P M > 0a:s. If M Hjj½A0 jj > 0 and jjE~c1 jj < 1, then
59
D.H. Hong / Information Sciences 228 (2013) 45–60
limt!1
E½NðtÞ 1 ¼ : t jjE~c1 jj
Proof. According to Theorem 12, we know that fNðxÞðtÞ=tg converges in credibility to 1=jjE~c1 jj. It follows from Liu [20, Theorem 3.54] that for any r 2 R,
limt!1 CrfNðxÞðtÞ=t P rg ¼ Cr
1 Pr jjE~c1 jj
a:s:;
and hence by Lebesgue dominated convergence theorem, we have
limt!1 ChfNðtÞ=t P rg ¼ limt!1
Z
CrfNðxÞðtÞ=t P rgdPðxÞ ¼ X
Z
Cr
X
1 1 P r dPðxÞ ¼ Ch Pr : jjE~c1 jj jjE~c1 jj
By Lemma 5, we note that, for large r > 0,
Ch
NðtÞ P r ¼ 0: t
As a consequence, by Lebesgue dominated convergence theorem, we have
limt!1
E½NðtÞ ¼ limt!1 t
Z
1
ChðNðtÞ=t P rÞdr ¼
0
1 : jjE~c1 jj
h Example 4. Let A be a fuzzy number on R2 defined by Aðx; yÞ ¼ maxf0; 1 ðx2 þ y2 Þg. Then ½A0 ¼ fðx; yÞ : x2 þ y2 6 1g. Let ~i ðxÞg1 fa i¼1 be a sequence of positive real valued i.i.d. random variables that are uniformly distributed on ½1; 3 and 2 f~ci ðxÞg1 i¼1 be a sequence of R -valued i.i.d. random variables that are uniformly distributed on ½3; 7 ½4; 10. Then ~ ~ Ec1 ¼ ð5; 7Þ; Ea1 ¼ 2; H ¼ 3; M ¼ 5, and jj½A0 jj ¼ 1. Let fnn g be a sequence of fuzzy random variables such that ~n ðxÞAðx ~cn ðxÞÞ. Then nn ðxÞðxÞ ¼ a
M Hjj½A0 jj ¼ 2 > 0: Then fnn g are positive fuzzy random variables and by Theorem 18, 19 and 20, we have, for any
SNðtÞ
P ¼ 0; limt!1 Ch
ð5; 7Þ
NðtÞ
> 0,
NðtÞ 1 pffiffiffiffiffiffi P ¼ 0 limn!1 Ch t 74 and
limt!1
E½NðtÞ 1 ¼ pffiffiffiffiffiffi : t 74
8. Conclusion This paper considers the law of large numbers and the renewal reward process in which the inter-arrival time is modeled as weighted fuzzy variables on Rp under continuous Archimedean t-norm-based fuzzy operations. The paper provides a far-reaching generalization of previous findings on the law of large numbers for T-related fuzzy numbers and derives different versions of the fuzzy renewal theorem for the rate of a renewal process having fuzzy inter-arrival times by using the law of large numbers for T-related weighted fuzzy variables. In addition, the paper considers the law of large numbers and the renewal process for Trelated fuzzy variables with respect to the credibility measure, the chance measure, and the expected value. Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0021089). References [1] R. Badard, The law of large numbers for fuzzy processes and the estimation problem, Information Science 28 (1982) 161–178. [2] G. Choquet, Theory of capacities Ann. Inst. Fourier, Grenoble 5 (1955) 131–295. [3] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets, World Scientific Publishing Co. Pte. Ltd., 1994.
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[4] D. Dubois, H. Prade, Linear programming with fuzzy data, in: J. Bezdek (Ed.), Analysis of Fuzzy Information, Application in Engineering and Science, vol. 3, CRC Press, Boca Raton, FL, 1987, pp. 241–261. [5] R. Fullér, A law of large numbers for fuzzy numbers, Fuzzy Sets and Systems 45 (1992) 299–303. [6] R. Fullér, E. Triesch, A note on the law of large numbers for fuzzy variables, Fuzzy Sets and Systems 55 (1993) 235–236. [7] R. Fullér, T. Keresztfalvi, On generalization of Nguyen’s theorem, Fuzzy Sets and Systems 41 (1991) 371–374. [8] D.H. Hong, P. Ro, The law of large numbers for fuzzy numbers with unbounded supports, Fuzzy Sets and Systems 116 (2000) 269–274. [9] D.H. Hong, J. Lee, On the law of large numbers for mutually T-related fuzzy numbers, Fuzzy Sets and Systems 121 (2001) 537–543. [10] D.H. Hong, C.H. Ahn, Equivalent conditions for laws of large numbers for T-related L–R fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 387–395. [11] D.H. Hong, Renewal process with T-related fuzzy inter-arrival times and fuzzy rewards, Information Sciences 176 (2006) 2386–2395. [12] D.H. Hong, Blackwell’s theorem for T-related fuzzy variables, Information Sciences 180 (2010) 1769–1777. [13] D.H. Hong, Uniform convergence of fuzzy random renewal process, Fuzzy Optimization and Decision Making 9 (2010) 275–288. [14] D.H. Hong, Renewal process for fuzzy variables, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15 (2007) 493–501. [15] C-M. Hwang, A theorem of renewal process for fuzzy random variables and its application, Fuzzy Sets and Systems 116 (2000) 237–244. [16] E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Trends in Logic, 8, Kluwer, Dordrecht, 2000. [17] H. Kwakernaak, Fuzzy random variables I: definitions and theorems, Information Sciences 15 (1978) 1–29. [18] Y.-K. Liu, B. Liu, Fuzzy random variable: a scalar expected value operator, Fuzzy Optimization and Decision Making 2 (2003) 143–160. [19] Y.-K. Liu, B. Liu, On minimum-risk problems in fuzzy random decision systems, Computers and Operations Research 32 (2005) 257–283. [20] B. Liu, Uncertain Theory: An Introduction to its Axiomatic Foundations, Springer-Verlag, Berlin, 2004. [21] B. Liu, Y.-K. Liu, Expected value of a fuzzy variable and fuzzy expected value models, IEEE Transaction on Fuzzy Systems 10 (2002) 445–450. ˘ anová, T-law of large numbers for fuzzy numbers, Kybernetika 36 (2000) 379–388. [22] A. Marková-Stupn [23] T. Murofushi, M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets Systems 29 (1989) 201–227. [24] T. Murofushi, M. Sugeno, A theory of fuzzy measure representations the Choquet integral and null set, Journal of Mathematical Analysis and Applications 159 (1991) 532–549. [25] T. Murofushi, M. Sugeno, M. Machida, Non-monotone fuzzy measure and the Choquet integral, Fuzzy Sets Systems 64 (1994) 73–86. [26] Y. Narukawa, T. Murofushi, M. Sugeno, Regular fuzzy measure and representation of comonotonically additive functional, Fuzzy Sets Systems 112 (2000) 177–186. [27] E. Popova, H.C. Wu, Renewal reward processes with fuzzy rewards and their applications to T-age replacement policies, European Journal of Operational Research 117 (1999) 606–617. [28] M.L. Puri, D.A. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Applications 114 (1986) 409–422. [29] E. Triesch, Characterization of Archimedean t-norms and a law of large numbers, Fuzzy Sets and Systems 58 (1993) 339–342. [30] S. Wang, Y-K. Liu, J. Watada, Fuzzy random renewal process with queueing applications, Computers and Mathematics with Applications 57 (2009) 1232–1248. [31] R.C. Williamson, The law of large numbers for fuzzy variables under a general triangular norm extension principle, Fuzzy Sets and Systems 41 (1991) 55–81. [32] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3–28. [33] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. [34] R. Zhao, B. Liu, Renewal process with fuzzy inter-arrival times and rewards, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11 (2003) 573–586. [35] R. Zhao, W. Tang, Some properties of fuzzy random processes, IEEE Transactions on Fuzzy Systems 2 (2006) 173–179. [36] R. Zhao, W. Tang, C. Wang, Fuzzy random renewal process and renewal reward process, Fuzzy Optimization and Decision Making 6 (2007) 279–295.
Contents lists available at SciVerse ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
The law of large numbers and renewal process for T-related weighted fuzzy numbers on Rp Dug Hun Hong Department of Mathematics, Myongji University, Yongin, Kyunggido 449-728, South Korea
a r t i c l e
i n f o
Article history: Received 28 April 2011 Received in revised form 19 October 2012 Accepted 9 December 2012 Available online 19 December 2012 Keywords: Fuzzy numbers Law of large numbers Continuous Archimedean t-norm Fuzzy renewal process Possibility measure Chance measure
a b s t r a c t This paper considers the law of large numbers for T-related weighted fuzzy variables whose underlying spaces are Rp . We provide new sufficient conditions that do not depend on the shape of fuzzy numbers and the additive generator of a t-norm T for the law of large numbers for T-related weighted fuzzy variables. The results provide a far-reaching generalization of previous findings on the law of large numbers for T-related fuzzy numbers. In addition, we prove T-related fuzzy renewal theorems in which the inter-arrival time is characterized as weighted fuzzy numbers under t-norm-based fuzzy operations on Rp by using the law of large numbers for weighted fuzzy variables on Rp . We also investigate the law of large numbers and the renewal process for T-related fuzzy numbers on Rp with respect to the credibility measure, the chance measure, and the expected value. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The theory of fuzzy sets, introduced by Zadeh [32,33], has been widely examined and applied to statistics and possibility theory in recent years. Since Puri and Ralescu’s [28] introduction of the concept of fuzzy random variables, there has been growing interest in fuzzy variables. A number of studies [11–15,27,30,34–36] have investigated renewal theory in the fuzzy random environment based on the concept of fuzzy variable and fuzzy random variable. All of these studies used min-normbased fuzzy operations. In general, we can consider the extension principle realized by the means of some t-norm. Renewal theory is closely related to the law of large numbers. In 1982, Badard [1] proved the law of large numbers for fuzzy numbers with a common spread when Tðu; v Þ ¼ uv . In 1992, Fullér [5] proved the law of large numbers for a sequence of mutually T-related symmetric triangular fuzzy numbers with a common spread when Tðu; v Þ 6 H0 ðu; v Þ ¼ uv =ðu þ v uv Þ for all 0 6 u; v 6 1. A number of studies have examined various types of the law of large numbers for T-related fuzzy variables, including Williamson [31], Fullér and Triesch [6], Triesch [29], Marková [22], Hong and Lee [9], Hong and Ro [8], and Hong and Ahn [10]. Recently, Hong [11] considered a renewal reward process in which the inter-arrival time and the reward are modeled as fuzzy variables by using t-norm-based fuzzy operations and proposed a fuzzy renewal theorem and a fuzzy renewal reward theorem through the necessity measure. In [12], Hong evaluated fuzzy versions of Blackwell’s theorem by considering the necessity measure and the expected value of fuzzy variables. Wang et al. [30] obtained some limit theorems by considering the chance measure and the expected value for the sum of fuzzy random variables on the basis of continuous Archimedean tnorm-based arithmetic operations. Most of the results were based on R. In this paper, we first examine the law of large numbers for T-related weighted fuzzy numbers on Rp by using new methods. The results generalize most of the previous findings on the law of large numbers for T-related fuzzy numbers on R. Based E-mail address: [email protected] 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.12.016
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D.H. Hong / Information Sciences 228 (2013) 45–60
on these results, we examine the fuzzy renewal process for T-related fuzzy numbers on Rp with respect to the chance measure and the expected value. In Section 2, we provide the definitions of the continuous t-norm, a representation of the Archimedean t-norm, and t-norm-based fuzzy operations on a separable Banach space. In Section 3, we discuss the law of large numbers for weighted fuzzy numbers on Rp . We provide two convergence theorems for T-related weighted fuzzy numbers on Rp with respect to the general continuous t-norm and the continuous Archimedean t-norm. In Section 4, we discuss the law of large numbers for T-related fuzzy numbers with random centers and random spreads. In Section 5, we discuss the renewal process by considering the fuzzy inter-arrival time and the fuzzy reward on Rp . We derive a fuzzy renewal theorem for the rate of a renewal process having fuzzy inter-arrival times by using the law of large numbers for fuzzy numbers. In Section 6, we consider the law of large numbers and the renewal process for T-related fuzzy numbers with respect to the credibility measure. In Section 7, we consider the law of large numbers and the renewal process for T-related fuzzy numbers with respect to the chance measure and the expected value. Finally, in Section 8, some conclusion are given. 2. Preliminaries Let S be a separable Banach space with the norm k k; KðSÞ denote the class of nonempty compact convex subsets of S, and F ðSÞ denote the class of upper semicontinuous elements V of ½0; 1S such that the a-level sets V a belong to KðSÞ for all a 2 ½0; 1 with V a ¼ fx 2 SjVðxÞ P ag for a 2 ð0; 1, and V 0 ¼ clfx 2 SjVðxÞ > 0g. The space KðSÞ can be endowed with a linear structure induced by the scalar product and the Minkowski addition, that is,
kA ¼ fkaja 2 Ag; A þ B ¼ fa þ bja 2 A; b 2 Bg; for all A; B 2 KðSÞ, and k 2 R. If dH is the Hausdorff metric on KðSÞ, which for A; B 2 KðSÞ is given by
dH ðA; BÞ ¼ max sup inf ka bk; sup inf ka bk ; a2A
b2B
b2B
a2A
then ðKðSÞ; dH Þ is a complete and separable metric space ([3]). The norm of an element of KðSÞ is denoted by
kAk ¼ dH ðA; f0gÞ ¼ supfkxkjx 2 Ag: On the other hand, F ðSÞ can be endowed with the metric d1 defined ([3]) as follows:
d1 ðV; WÞ ¼ sup dH ðV a ; W a Þ;
for all V; W 2 F ðSÞ:
a2ð0;1
Here ðF ðSÞ; d1 Þ is a complete metric space ([3]). Recall that a triangular norm (or a t-norm) is a commutative monoid operation in ½0; 1 with neutral element 1 and is monotonic (non-decreasing) when viewed as a bivariate function. Now suppose that Ai 2 F ðSÞ; i ¼ 1; . . . ; n and a t-norm T ([16]) are given. Then the T-sum A1 þ þ An 2 F ðSÞ is defined via the sup-T extension principle by
ðA1 þ þ An ÞðzÞ ¼
sup TðA1 ðx1 Þ; . . . ; An ðxn ÞÞ; z 2 S: x1 þþxn ¼z
Following Fullér and Keresztfalvi [7], if T is upper semicontinuous then the equality
½A1 þ þ An a ¼
[
½A1 a1 þ þ ½An an ; a 2 ð0; 1
ð1Þ
Tða1 ;...;an ÞPa
holds. For the special case in which T ¼ min, the addition is denoted by . Then it holds that
ðkVÞa ¼ kV a ; ½A1 An a ¼ ½A1 a þ þ ½An a ;
ð2Þ
for all k 2 R, and a 2 ½0; 1 [4]. Recall that a continuous t–norm T is called Archimedean if and only if Tðx; xÞ < x for all x 2 ð0; 1Þ. A well-known theorem ([16]) asserts that for each continuous Archimedean t-norm there exists a continuous, decreasing function f : ½0; 1 ! ½0; 1 with f ð1Þ ¼ 0 such that
Tðx1 ; . . . ; xn Þ ¼ f ½1 ðf ðx1 Þ þ þ f ðxn ÞÞ for all xi 2 ð0; 1Þ; 1 6 i 6 n. Here f ½1 : ½0; 1 ! ½0; 1 is defined by
( f
½1
ðyÞ ¼
f 1 ðyÞ for y 2 ½0; f ð0Þ; 0
if y > f ð0Þ:
The function f is called the additive generator of T. Note that if a continuous t-norm T has an additive generator f, then this additive generator is uniquely determined up to a non-zero positive multiplicative constant.
D.H. Hong / Information Sciences 228 (2013) 45–60
47
Because f is continuous and decreasing, f ½1 is also continuous and non-increasing, we have from (1)
[
½A1 þ þ An a ¼
½A1 a1 þ þ ½An an :
ð3Þ
f ða1 Þþþf ðan Þ6f ðaÞ
Example 1. Examples of continuous Archimedean t-norms with additive generators are listed as follows: (a) Dombi t-norm D; p > 0:
Dp ðx; yÞ ¼
1 1
1 þ ðð1x 1Þp þ ð1y 1Þp Þp
with the additive generator f ðxÞ ¼ ðð1 xÞ=xÞp . (b) Hamacher t-norm H; p P 0:
Hp ðx; yÞ ¼
xy p þ ð1 pÞðx þ y xyÞ
with the additive generator f ðxÞ ¼ lnððp þ ð1 pÞxÞ=xÞ. (c) Sklar t-norm S; p > 0: 1
Sp ðx; yÞ ¼ ðxp þ yp 1Þ p
with the additive generator f ðxÞ ¼ ð1=pÞðxp 1Þ. (d) Frank t-norm F; p > 0; p – 1:
ðpx 1Þðpy 1Þ F p ðx; yÞ ¼ log p 1 þ p1 with the additive generator f ðxÞ ¼ log p ððp 1Þ=ðpx 1ÞÞ. From a representation theorem in topological semigroup theory [4], every continuous t-norm is the ordinal sum of a family of Archimedean t-norms. Namely, there exists a family of subintervals f½mi ; ni gi2I having non-overlapping interiors in ½0; 1 such that the following assertions hold: (i) The restriction of T to ½mi ; ni ½mi ; ni is T i Ui , where T i is an Archimedean t-norm and /i is the natural homomorphism
Ui : ðx; yÞ 2 ½mi ; ni ½mi ; ni #
x mi y mi ; ni ni ni mi
¼ ð/i ðxÞ; /i ðyÞÞ
and 0=0 ¼ 0 by convention. (ii) Elsewhere, T is the minimum. Because there are at most countably many subintervals for which ai – bi ; ½0; 1 can be decomposed into a countable union of closed subintervals with non-overlapping and non-empty interiors where either ‘T is Archimedean’ or ‘T ¼ min’ in the sense of (i). We denote KA for A 2 F ðSÞ by ½KAa ¼ ½Ani if a 2 ½mi ; ni for some i, ½KAa ¼ ½Aa , otherwise. A fuzzy variable (or a fuzzy set) on Rp is a function from Rp to ½0; 1. Definition 1 [5]. A function A 2 F ðRp Þ is called a fuzzy number (a special type of fuzzy variable) on Rp if there exists a unique element m0 2 Rp satisfying Aðm0 Þ ¼ supx AðxÞ ¼ 1. The number m0 ¼ mðAÞ is called the modal value of A. 3. The law of large numbers for weighted fuzzy numbers on Rp Throughout this section, let S ¼ Rp with the Euclidean norm k k. Let A be a fuzzy number on Rp and a 2 R1 . Then aA is a fuzzy number on Rp such that
aAðxÞ ¼ A
x ; thatis; ½aAa ¼ a½Aa ; a 2 ½0; 1: a
In addition, if B is a fuzzy number on Rp and c 2 Rp ; a > 0 such that BðxÞ ¼ aAðx cÞ, then
½Ba ¼ ½aAa þ c ¼ a½Aa þ c; a 2 ½0; 1: Let fbn g be a sequence of real numbers and fBn g be a sequence of fuzzy numbers. Then fbn Bn g is called a sequence of weighted fuzzy numbers.
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D.H. Hong / Information Sciences 228 (2013) 45–60
For a fuzzy number n and any subset D Rp , the quantity
Nesfn 2 Dg :¼ 1 supnðxÞ :¼ 1 Posfn 2 Dc g xRD
is considered to measure the necessity of n belonging to D ([29]). Note that
Nesfn 2 Dg 6 Posfn 2 Dg: Following Fuller [5], we say that a sequence of fuzzy numbers fAn g obeys the law of large numbers if for all
Sn mðSn Þ
< ¼ limn!1 NesfðSn =n 2 BðmðSn Þ=n; Þg ¼ 1 limn!1 Nes
n
>0
where Bða; Þ ¼ fx 2 Sjka xk < g and Sn ¼ A1 þ A2 þ þ An . We note that if n be a fuzzy number with mðnÞ ¼ 0, then
Nesfknk < g ¼ Nesfn 2 Bð0; Þg > 1 a () Posfn 2 Bð0; Þc g < a () dH ðf0g; ½na Þ ¼ k½na k < : The following result is immediate from this fact. Proposition 1. A sequence of fuzzy numbers fAn g obeys the law of large numbers if and only if for any 0 < a < 1
limn!1 k½ðSn mðSn ÞÞ=na k ¼ 0 The following result is known. Proposition 2 [3]. Let A 2 F ðRp Þ and write C a ¼ ½Aa for a 2 ½0; 1. Then
dH ðC b ; C a Þ ! 0asb " a: We first introduce the following sufficient condition of the law of large numbers for fuzzy variables by using Proposition 1. Theorem 1. Let A be a fuzzy number on Rp with mðAÞ ¼ 0; An be a fuzzy number such that An ðxÞ ¼ an Aðx cn Þ, where cn 2 Rp ; an > 0; n ¼ 1; 2; . . ., denotes a sequence of fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm with P the additive generator f is given, limsupð ni¼1 ai =nÞ 6 M, for some M > 0, and limðmax16i6n ai =nÞ ¼ 0. Then we have, for all > 0
Sn mðSn Þ
< ¼1 limn!1 Nes
n
P Proof. Let 0 < a < 1; > 0 be given and f ð1 Þ ¼ t 0 . Suppose that ni¼1 f ðai Þ 6 f ðaÞ and let Ha; ¼ fkjf ðak Þ > t 0 g. Then the number of Ha; is less than or equal to N where N is the smallest natural number bigger than f ðaÞ=t 0 . We note that if k R Ha; , then ak P 1 . Then, we have from (3)
½a1 A þ þ an Aa ¼
[ Pn i¼1
ð½a1 Aa1 þ þ ½an Aan Þ
f ðai Þ6f ðaÞ
½a1 A1 þ þ ½an A1 þ
[ Pn i¼1
f ðxi Þ6f ðaÞ
¼ a1 ½A1 þ þ an ½A1 þ NBð0; tn Þ ¼
[ Pn 0 @
i¼1
f ðai Þ6f ðaÞ
X
0 @
X
½ak A1 þ
kRHa;
X
1 ½ak A0 A
k2Ha;
1
½ak A0 A ½a1 A1 þ þ ½an A1 þ NBð0; tn Þ
k2Ha;
X
ai ½A1 þ NBð0; t n Þ;
16i6n
where t n ¼ max16i6n ai k½A0 k and the second equality above comes from the convexity of ½A0 . Hence, we have, for large n,
Pn
Sn mðSn Þ 1
¼ ½a1 A þ þ an A 6 i¼1 ai k½A k þ N B 0; t n 6 Mk½A k þ N t n : a
1 1
n n n n n a
The first term goes to 0 because is arbitrary and lim!0 k½A1 k ¼ 0 by Proposition 2 and the second term goes to 0 as n ! 1 by assumption. This completes the proof. P In Theorem 1, if fan g is bounded, that is, an 6 K; n 2 N, for some K > 0, then two conditions limsupð ni¼1 ai =nÞ 6 M, for some M > 0, and limðmax16i6n ai =nÞ ¼ 0 are satisfied. Thus, the following result is immediate. h
D.H. Hong / Information Sciences 228 (2013) 45–60
49
Corollary 1. Let A be a fuzzy variable on Rp with mðAÞ ¼ 0 and let An be a fuzzy variable such that An ðxÞ ¼ an Aðx cn Þ, where cn 2 Rp ; an > 0; n ¼ 1; 2; . . ., denotes a sequence of fuzzy variables on Rp . Suppose that a continuous Archimedean t-norm with additive generator f is given and that fan g is bounded, that is, an 6 K; n 2 N, for some K > 0. Then the sequence fAn g obeys the law of large numbers. Note 1. One of the most generalized result of a law of large numbers for fuzzy numbers of Fullér [6] was proved by Marková [22, Theorem 2]. This is the case of fuzzy variables on R1 with cn ¼ 0; an 6 K for all n ¼ 1; 2; . . ., in Corollary 1. We now consider a necessary condition of the law of large numbers for weighted fuzzy numbers.
Theorem 2. Let A be a fuzzy number on Rp with mðAÞ ¼ 0 and let An be a fuzzy number such that An ðxÞ ¼ an Aðx cn Þ, where cn 2 Rp ; an > 0; n ¼ 1; 2; . . ., denotes a sequence of fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm with an additive generator f is given and that the sequence fAn g obeys the law of large numbers. Then we have
limðmax16i6n ai =nÞ ¼ 0: Proof. We have from (3)
½a1 A þ þ an Aa ¼
[ Pn i¼1
ð½a1 Aa1 þ þ ½an Aan Þ ¼
f ðai Þ6f ðaÞ
[ Pn i¼1
ða1 ½Aa1 þ þ an ½Aan Þ:
f ðai Þ6f ðaÞ
Let max16i6n ai ¼ an and choose an ¼ a, and ai ¼ 1; i – n . Then we have
Pn
a ¼ f ðaÞ and
i¼1 f ð i Þ
½a1 A þ þ an Aa an ½Aan ¼ max16i6n ai ½Aa ; and hence
Sn mðSn Þ 1
¼ ½a1 A þ þ an A P max16i6n ai k½A k: a
a
n n n a Therefore the result follows from the assumption. h Combining Theorems 1 and 2, we have the following result. Theorem 3. Let A be a fuzzy number on Rp with mðAÞ ¼ 0 and let An be a fuzzy number such that An ðxÞ ¼ an Aðx cn Þ, where cn 2 Rp ; an > 0; n ¼ 1; 2; . . ., denotes a sequence of fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm with an P additive generator f is given and that limsupð ni¼1 ai =nÞ 6 M, for some M > 0. Then the sequence of fuzzy numbers fAn g obeys the law of large numbers if and only if limðmax16i6n ai =nÞ ¼ 0.
Note 2. This is a different type of equivalent condition from Hong and Ahn [10, Theorem 3] for obeying the law of large numbers for fuzzy numbers. The conditions have nothing to do with the shape of fuzzy numbers and the additive generator of the continuous Archimedean t-norm. The following result shows the convergence of sequences of fuzzy variables with respect to a general continuous t-norm. From this result, we can see that the Archimedean property is essential for the law of large numbers. We need the following lemma which is easy to prove.
Lemma 1. Let Ai 2 Rp ; i ¼ 1; 2; 3 such that A1 A2 A3 . Then we have
dH ðA1 ; A2 Þ 6 dH ðA1 ; A3 Þ: Theorem 4. Let A be a fuzzy variable on Rp with mðAÞ ¼ 0; An ¼ an A; n ¼ 1; 2; . . ., where A 2 F ðRp Þ; an > 0, and T be a continuous P t-norm. Suppose that limð ni¼1 ai =nÞ ¼ 1 and limðmax16i6n ai =nÞ ¼ 0. Then we have for all 0 < a 6 1,
lim dH
n!1
Sn ; ½KAa ¼ 0: n a
Proof. Let T be a continuous t-norm such that the restriction of T to ½mi ; ni ½mi ; ni is indicated by T i /i , where T i is an Archimedean t-norm with an additive generator fi . If a R [ðmi ; ni Þ, because T ¼ min except on [fðmi ; ni Þ ðmi ; ni Þg, then we clearly have
50
D.H. Hong / Information Sciences 228 (2013) 45–60
[
½a1 A þ þ an Aa ¼
! n X ai ½Aa ;
ð½a1 Aa1 þ þ ½an Aan Þ ¼ ½a1 Aa þ þ ½an Aa ¼
Tða1 ;...;an ÞPa
i¼1
and hence the result follows immediately. Now, let mj < a < nj for some j. Note that if Tða1 ; . . . ; an Þ P a, then there exists ða01 ; . . . ; a0n Þ such that ai P a0i ; mj 6 a0i 6 nj ; i ¼ 1; . . . ; n and Tða01 ; . . . ; a0n Þ P a. Then, because ½ai Aai ½ai Aa0 ; i ¼ 1; . . . ; n, we have i
[
½a1 A þ þ an Aa ¼
ð½a1 Aa1 þ þ ½an Aan Þ
Tða1 ;...;an ÞPa
¼
[
Pn
[ T j ða01 ;...;a0n ÞPa
ð½a1 Aa0 þ þ ½an Aa0n Þ 1
ð½a1 Aa0 þ þ ½an Aa0n Þ
ð4Þ
1
l¼1
fj ð/j ða0 ÞÞ6fj ð/j ðaÞÞ l
Let > 0 be a number such that nj > mj and let fj ð/j ðnj ÞÞ ¼ t0 ; Ha; ¼ fkjfj ð/j ða0k ÞÞ > t0 g. Then the number of Ha; is less than N where N is the smallest natural number bigger than fj ð/j ðaÞÞ=t 0 . Note that if k R Ha; , then a0k P nj . Then, we have from (4) that
[
½a1 A þ þ an Aa ¼
ð½a1 Aa1 þ þ ½an Aan Þ
Tða1 ;...;an ÞPa
[ Pn
0 @
f ð/ ða0 ÞÞ6fj ð/j ðaÞÞ l¼1 j j l
[
ð½a1 Aa0 þ þ ½an Aa0n Þ
Pn
1
f ð/ ða0 ÞÞ6fj ð/j ðaÞÞ l¼1 j j l
X
½ak Anj þ
kRHa;
½a1 Anj þ þ ½an Anj þ
X
1
½ak A0 A
k2Ha;
[ Pn l¼1
0 @
fj ð/j ða0l ÞÞ6fj ð/j ðaÞÞ
X
1 ½ak A0 A
k2Ha;
½a1 Anj þ þ ½an Anj þ NBð0; t n Þ ¼ a1 ½Anj þ þ an ½Anj þ NBð0; t n Þ
n X ak ½Anj þ NBð0; tn Þ;
ð5Þ
k¼1
where t n ¼ max16k6n ak k½A0 k. On the other hand,
½a1 A þ þ an Aa ½a1 A þ þ an Anj ¼ ½a1 Anj þ þ ½an Anj ¼
n X
ak ½Anj :
ð6Þ
k¼1
Then
dH
Pn Pn Pn Pn Sn k¼1 ak k¼1 ak k¼1 ak k¼1 ak ; ½KAa ¼ dH ½Aa ; ½Anj 6 dH ½Anj ; ½Anj þ dH ½Anj ; ½Aa n a n n n n Pn Pn Pn a a a t n k k k k¼1 k¼1 k¼1 ½Anj ; ½Anj þ dH ½Anj ; ½Anj þ NB 0; 6 dH n n n n Pn Pn Pn a a a t n k¼1 k k¼1 k k¼1 k ½Anj ; ½Anj þ dH ½Anj ; ½Anj þ N 6 dH n n n n Pn Pn ak tn k¼1 ak dH ð½Anj ; ½Anj Þ þ N ; 6 k¼1 1 k½A0 k þ n n n
where the second inequality comes from (5), (6) and Lemma 1 and the third inequality comes from Proposition 2.4.1 [4]. The first and third terms go to 0 as n ! 1 by assumption, and the second term goes to 0 since is arbitrary and lim!0 dH ð½Ani ; ½Ani Þ ¼ 0 by Proposition 2, which completes the proof. h Note 3. The case of fuzzy variables on R1 with an ¼ 1 for all n ¼ 1; 2; . . ., which was proved by Marková ([22] Theorem 5) is a special case of Theorem 4.
4. The law of large numbers with random spreads and random centers e : X ! F ðRp Þ is a fuzzy random variable [16] if every a-cut of Let ðX; T ; PÞ be a probability space. The mapping Y p e e e Y ; Y a ðxÞ ¼ fx 2 R : ð Y ðxÞÞðxÞ P ag is a compact random set. For general discussions on fuzzy random variables, see Kwakernaak [17], Puri and Ralescu [28]. Let A 2 F ðRp Þ with ~ be a positive real valued random variable, ~c be Rp -valued random variable on ðX; T ; PÞ, and g : X ! F ðRp Þ be a mðAÞ ¼ 0; a ~ðxÞAðx ~cðxÞÞ. Then g is clearly a fuzzy random variable. function defined by gðxÞðxÞ ¼ a
D.H. Hong / Information Sciences 228 (2013) 45–60
51
Let A be a fuzzy number on Rp with mðAÞ ¼ 0. If B is a fuzzy variable on Rp and c ¼ ðc1 ; c2 ; . . . ; cp Þ 2 Rp ; a > 0 such that BðxÞ ¼ aAðx cÞ, then c is called the center of B and a is called the spread of B. In this section, we consider laws of large numbers for fuzzy numbers on Rp with random centers and spreads. The following important lemma is well known in probability theory. Lemma 2. Let fX n g1 and identically distributed (i.i.d.) random variables. Then EjX 1 j < 1 iff n¼1 be independent P n1 max16k6n jX k j ! 0 a.s. iff lim n1 nk¼1 X k ¼ E½X 1 a.s. By Lemma 2, the following is immediate from Theorem 1. 1 p ~i ðxÞg1 ~ Theorem 5. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued i.i.d. random variables, A be a fuzzy number on Rp with mðAÞ ¼ 0, and fAn g be a sequence of fuzzy random variables such that ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given. Then Ea ~1 < 1 if and only if the sequence An ðxÞðxÞ ¼ a of fuzzy numbers fAn g obeys the law of large numbers a.s., that is, for all > 0
Pn
Sn ðxÞ
~ i¼1 c i ðxÞ
< ¼ 1 a:s: limn!1 Nes
n
n We also obtain the following result if we follow the idea behind the proof in Theorem 6 [10] by using Theorem 5 and Lemma 2. 1 p ~i ðxÞg1 ~ Theorem 6. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued i.i.d. p random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fAn g be a sequence of fuzzy random variables such that ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given. Then Ea ~1 < 1 and Ejj~c1 jj < 1 if and only An ðxÞðxÞ ¼ a if for all > 0,
Sn ðxÞ
< ¼ 1 a:s: ~ limn!1 Nes
E c 1
n
~i ðxÞg1 Example 2. Let A be a fuzzy number on R2 with Aðx; yÞ ¼ maxf0; 1 ðx2 þ y2 Þg; fa i¼1 be a sequence of positive real val2 ued i.i.d. random variables which are uniformly distributed on ð0; 100, and f~ci ðxÞg1 i¼1 be a sequence of R -valued i.i.d. ran~1 < 1. Let fAn g be a sequence of fuzzy dom variables that are uniformly distributed on ½1; 3 ½2; 8. Then E~c1 ¼ ð2; 5Þ and Ea ~n ðxÞAðx ~cn ðxÞÞ. Then by Theorem 6, for any > 0, we have random variables such that An ðxÞðxÞ ¼ a
Sn ðxÞ
< ¼ 1 a:s: ð2; 5Þ limn!1 Nes
n 5. Renewal process with fuzzy inter-arrival times on Rp In this section, we assume that S ¼ Rp and consider the renewal process with fuzzy inter-arrival times. Let nn denote pdimensional times between the ðn 1Þth and nth events which are known as pdimensional inter-arrival times, n ¼ 1; 2; . . .. Define
S0 ¼ 0; Sn ¼ n1 þ n2 þ þ nn ; n P 1: If pdimensional inter-arrival times nn ; n ¼ 1; 2; . . . are fuzzy variables on Rp , then the process fnn ; n P 1g is called a pdimensional fuzzy renewal process. We assume that ½ni 0 ðRþ Þp , that is, ni is positive, i ¼ 1; 2; . . .. Let NðtÞ denote the total number of events that occurred by time t. Then we have
NðtÞ ¼ maxnP0 fnj0 < kSn k 6 tg; where kSn kðzÞ ¼ supkxk¼z Sn ðxÞ. It is clear that NðtÞ is a nonnegative integer valued fuzzy variable, and
PosfNðtÞ ¼ ng ¼ PosfkSn k 6 t < kSnþ1 kg; ( PosfNðtÞ < ng ¼ Pos
n1 [
ð7Þ
) fNðtÞ ¼ ig
¼ PosfkSn k > tg;
ð8Þ
¼ PosfkSn k 6 tg;
ð9Þ
i¼0
PosfNðtÞ P ng ¼ Pos
( 1 [
) fNðtÞ ¼ ig
i¼n
for each integer n. We call NðtÞ the fuzzy renewal variable. The following lemma is easy to check.
52
D.H. Hong / Information Sciences 228 (2013) 45–60
Lemma 3. Let fni g be a sequence of fuzzy variables on Rp obeying the law of large numbers. If mðSn Þ=n ! c as n ! 1, then for any > 0,
Sn
< ¼ 1: limn!1 Nes
c
n Theorem 7 (Fuzzy Renewal Theorem in Necessity). Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ ai Aðx ci Þ; i ¼ 1; 2; . . . be a sequence of positive fuzzy variables on Rp , and NðtÞ be a fuzzy renewal variable. Suppose that a continuous Archimedean tP norm is given, that limsupð ni¼1 ai =nÞ 6 K, for some K 2 R, and limðmax16i6n ai =nÞ ¼ 0. If mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1, then for any > 0,
NðtÞ 1 < ¼1 limt!1 Nes t kck Proof. Because ni ; i ¼ 1; 2; . . ., obeys the law of large numbers by Theorem 1, and mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1, we have by Lemma 3, for any > 0,
Sn
< ¼ 1: c limn!1 Nes
n
Noting that
Sn
jkSn k nkckj
< ; Nes
n c < 6 Nes n we have
limn!1 Nes
jkSn k nkckj < ¼ 1: n
Note that it suffices to prove that for small
ð10Þ
> 0,
NðtÞ 1 < limt!1 Nes ¼1 t kck Because NðtÞ is a fuzzy variable taking an integer value, we have
NðtÞ 1 NðtÞ 1 NðtÞ 1 < P 1 Pos _ Pos Nes < P þ t kck t kck 2 t kck 8 9 < = ik > t ¼ 1 Pos kSh 1 t : ; kck 2 8 9 < = i k 6 t ðwhere ½ is the smallest integer greater than or equal to Þ _ Pos kSh 1 þ t : ; kck 8 h 9 8 h 9 ik ik kS kS > > > > 1 t 1 þ t < = < = kck 2 kck t t i > h i _ Pos h i 6 h i ¼ 1 Pos h > > 1 1 1 1 : kck ; : kck ; 2 t 2 t > þ t þ t > kck kck where the first equality comes from (8) and (9). Then, because
limt!1 h
1 kck
t
1 i¼ 1 > kck; ð 2Þ t kck 2
and there exists an
limt!1 h
0 > 0 such that
1 > kck þ 0 1 ðkck 2Þ we have, for large t,
and
1 < kck 0 ; 1 ðkck þ Þ
t
1 kck
1 i¼ 1 < kck; ð þ Þ þ t kck
D.H. Hong / Information Sciences 228 (2013) 45–60
53
8 h 9 8 h 9 ik ik kS kS > > > > 1 t 1 þ t < = < = NðtÞ kck 2 kck 1 0 0 h i h i < > kck þ < kck P 1 Pos Nes _ Pos > > > > 1 t kck : ; : 1 þ t ; t kck
2
kck
kSn k kSm k R ½kck 0 ; kck þ 0 _ Pos R ½kck 0 ; kck þ 0 : P 1 Pos n m 1 1 þ tas mÞ tas n and ½ ðregarding½ kck 2 kck
Because
kSn k jkSn k nkckj R ½kck 0 ; kck þ 0 ¼ 1 Nes < 0 ; Pos n n goes to 0 as t ! 1 by (10), and similarly, PosfkSm k=m R ½kck 0 ; kak þ 0 g goes to 0 as t ! 1. Hence the proof is complete. h The following result is immediate from Theorem 7. Corollary 2. Let A be a fuzzy number on Rp with mðAÞ ¼ 0 and ni ðxÞ ¼ Aðx ci Þ; i ¼ 1; 2; . . . be a sequence of positive fuzzy variables on Rp . Suppose that a continuous Archimedean t-norm is given and that mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then we have, for any > 0,
NðtÞ 1 < ¼ 1: limt!1 Nes t kck Theorem 8. Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ ai Aðx ci Þ; i ¼ 1; 2; . . . be a sequence positive fuzzy variables on Rp obeying the law of large numbers, and NðtÞ be a fuzzy renewal variable. If mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1, then for any > 0,
SNðtÞ
< ¼ 1: limt!1 Nes
c
NðtÞ
Proof. Because it suffices to prove that for small
>0
SNðtÞ
limt!1 Nes
NðtÞ c < ¼ 1; we assume that 0 < < 1=kck. Then we have
SNðtÞ
< ¼ 1 Pos SNðtÞ c P ¼ 1 supn Pos SNðtÞ c P ; NðtÞ ¼ n Nes
c
NðtÞ
NðtÞ
NðtÞ
i Pos Sn c P ; NðtÞ ¼ n ¼ 1 sup h
n
1 t; 1 þ t n2 kck
kck
i Pos Sn c P ; NðtÞ ¼ n
1 t; 1 þ t n nR kck kck
i Pos Sn c P _ sup h i PosfNðtÞ ¼ ng P 1 sup h
1 t; 1 þ t 1 t; 1 þ t n n2 nR kck kck kck kck _ sup h
¼ 1 At _ Bt ; where
NðtÞ 1 1 1 < ¼ 1 Nes ! 0ast ! 1ðby Theorem7Þ t; þ t Bt ¼ Pos NðtÞ R kck kck t kck and
At ¼ sup h n2
1 kck
t;
i Pos Sn c P ¼ 1 inf h i Nes Sn c < ! 0ast
1 þ t 1 t; 1 þ t n n n2
kck
! 1ðby Lemma3Þ which completes the proof. h
kck
kck
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D.H. Hong / Information Sciences 228 (2013) 45–60
6. Results from the credibility measure In this section, we consider the law of large numbers and the renewal process for T-related fuzzy numbers with respect to the credibility measure. The credibility of n belonging to D is defined as
Crfn 2 Dg ¼ ð1=2ÞðPosfn 2 Dg þ Necfn 2 DgÞ: Before we present the expected value of a fuzzy variable, we first recall the definition of Choquet integral. This kind of integral was first introduced in [2], and later was restudied in the field of fuzzy measure theory by some researchers such as Murofushi and Sugeno [23,24], Murofushi, Sugeno and Machida [25], and Narukawa, Murofushi and Sugeno [26]. Let ðX; B; lÞ be a fuzzy measure space. The Choquet integral of a nonnegative measurable function with respect to a fuzzy measure l is defined as
ðCÞ
Z
fdl ¼
Z
1
lðfxjf ðxÞ P rgÞdr 0
If
l is a finite fuzzy measure, then the Choquet integral of a measurable function with respect to l is defined as ðCÞ
Z
fdl ¼ ðCÞ
Z
f þ dl ðCÞ
Z
f dld
where f þ ¼ f _ 0; f ¼ ðf ^ 0Þ, and ld is the dual of l in the sense that lðAÞ ¼ lðXÞ ld ðAC Þ. And from the measure-theoretic interpretation of Choquet integral, it is usually regarded as the generalization of usual mathematical expectation. Therefore, motivated by the idea of Choquet integral, E½n(see [21]) is defined as
E½n ¼
Z
1
Crðn P rÞdr
0
Z
0
Crðn 6 rÞdr 1
provided that at least one of the two integrals is finite. In particular, if n is a nonnegative fuzzy variable (i.e., Crfn < 0g ¼ 0), R1 then E½n ¼ 0 Crfn P rgdr. Definition 2. The sequence fnn g converges in necessity to n if for any
> 0,
limn!1 Necfknn nk < g ¼ 1: Definition 3. The sequence fnn g converges in credibility to n if for any
> 0,
limn!1 Crfknn nk P g ¼ 0: Note that the condition
lim Nesfnn 2 Dn g ¼ 1
n!1
implies
Crfnn 2 Dcn g 6 Posfnn 2 Dcn g ¼ 1 Nesfnn 2 Dn g ! 0: Hence we have the following result. Proposition 3. If the sequence fnn g converges in necessity to n, then fnn g converges in credibility to n. By Lemma 3 and Proposition 3, the following result is immediate from Theorem 1. Theorem 9. Let A be a fuzzy number on Rp with mðAÞ ¼ 0, and nn ðxÞ ¼ an Aðx cn Þ; cn 2 Rp ; an > 0; n ¼ 1; 2; . . . denote a sequence P of fuzzy variables on Rp . Suppose that a continuous Archimedean t-norm is given, limsupð ni¼1 an =nÞ 6 K, for some K 2 R; limðmax16i6n ai =nÞ ¼ 0, and mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then we have, for any > 0,
Sn
P ¼ 0: limn!1 Cr
c
n
By Proposition 3, the following is immediate from Theorem 7. Theorem 10. Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ ai Aðx ci Þ; n ¼ 1; 2; . . . be a sequence of positive fuzzy variP ables on Rp , and NðtÞ be fuzzy renewal variable. Suppose that a continuous Archimedean t-norm is given, limsupð ni¼1 an =nÞ 6 K, for some K 2 R; limðmax16i6n ai =nÞ ¼ 0, and limn!1 ðc1 þ þ cn Þ=n ¼ c. Then
D.H. Hong / Information Sciences 228 (2013) 45–60
55
NðtÞ 1 P ¼ 0: limn!1 Cr t kck The following result is immediate from Theorem 10. Corollary 3. Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ Aðx ci Þ; n ¼ 1; 2; . . . be a sequence of positive fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm is given and that mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then fNðtÞ=tg con1 verges in credibility to kck . Lemma 4. Let A be a fuzzy number on Rp with mðAÞ ¼ 0, and ni ðxÞ ¼ ai Aðx ci Þ; i ¼ 1; 2; . . . be a sequence of positive fuzzy numbers on Rp . Suppose that a continuous Archimedean t-norm is given, ða1 þ þ an Þ=n ! a as n ! 1, and mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then for large r > 0, and for large t > 0, we have
Cr
NðtÞ P r ¼ 0: t
Proof. Because ni ðxÞ ¼ ai Aðx ci Þ; i ¼ 1; 2; . . . are positive fuzzy variables, limn!1 ða1 þ þ an Þ=n ¼ a, and limn!1 ðc1 þ þ cn Þ=n ¼ c, there exists 1 ; 2 > 0 such that 0 < 2 ¼ inf fjjxjj : x 2 ða þ 1 Þ½A0 þ cg. Let I be an indicator function. We consider that
S½rt
NðtÞ
6 t=½rt 6 Pos S½rt 6 t P r ¼ PosfNðtÞ P rtg ¼ PosfNðtÞ P ½rtg ¼ PosfkS½rt k 6 tg ¼ Pos
Pos
t tr ½rt ½rt 8 ½rt 9 X > > > > > > Iai ½A0 þci ðzÞ > >
<
S½rt 1 S½rt 1 1= i¼1
¼ Pos 6 ¼ sup 6 sup ðzÞ : kzk 6 : kzk 6 > r r r> ½rt ½rt ½rt > > > > > > : ; where ½rt is the smallest integer such that ½rt P rt. Because, for large t > 0
P½rt
i¼1 I ai ½A0 þci ðzÞ
½rt
¼ IP½rt a
P½rt ðzÞ 6 Iðaþ1 Þ½A0 þc ðzÞ c
i¼1 i ½A þ 0 ½rt
i¼1 i ½rt
and
2 sup Iðaþ1 Þ½A0 þc ðzÞ : kzk 6 ¼ 0; 2 we have, for large t > 0 and for r > 2=2 ,
NðtÞ P r ¼ 0; Pos t which proves the result noting that
Cr
NðtÞ NðtÞ P r 6 Pos Pr : t t
h Theorem 11 (Fuzzy Renewal Theorem in Credibility). Let A be a fuzzy number on Rp with mðAÞ ¼ 0; ni ðxÞ ¼ ai Aðx ci Þ, be a sequence of positive fuzzy variables on Rp , and NðtÞ be fuzzy renewal variable. Suppose that a continuous Archimedean t-norm given, limðmax16i6n ai =nÞ ¼ 0; ða1 þ þ an Þ=n ! a as n ! 1, and mn ¼ ðc1 þ þ cn Þ=n ! c as n ! 1. Then
limt!1
E½NðtÞ 1 ¼ : t kck
Proof. According to Theorems 10, we know that fNðtÞ=tg converges in credibility to 1=kck. Then, it follows from Liu [20, Theorem 3.54] that for any r 2 R,
limt!1 CrfNðtÞ=t P rg ¼ Cr
1 Pr : kck
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D.H. Hong / Information Sciences 228 (2013) 45–60
By Lemma 4, we note that, for large t > 0 and for r > 2=,
Cr
NðtÞ NðtÞ P r 6 Pos P r ¼ 0: t t
As a consequence, by Lebesgue dominated convergence theorem, we have
limt!1
E½NðtÞ ¼ limt!1 t
Z
1
CrfNðtÞ=t P rgdr ¼
0
1 : kck
By Proposition 3 and Lemma 2, the following is immediate from Theorem 7. h 1 p ~i ðxÞg1 ~ Theorem 12. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given. If Ea ~1 < 1, and Ejj~c1 jj < 1, then that nn ðxÞðxÞ ¼ a
NðxÞðtÞ 1 limt!1 Cr t jjE~c
1
P ¼ 0 a:s: jj
By Lemma 2, the following is immediate from Theorem 11. ~i ðxÞg1 Theorem 13 (Fuzzy Random Renewal Theorem in Credibility). Let fa i¼1 be a sequence of positive real valued i.i.d. random p variables, f~ci ðxÞg1 be a sequence of R -valued i.i.d. random variables, A be a fuzzy number on Rp with mðAÞ ¼ 0, and fnn g be a i¼1 ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean tsequence of positive fuzzy random variables such that nn ðxÞðxÞ ¼ a ~1 < 1, and Ejj~c1 jj < 1. Then we have norm is given, Ea
limt!1
E½NðxÞðtÞ 1 ¼ a:s: t jjE~c1 jj
By Proposition 3, the following is immediate from Theorem 8. Theorem 14. Let A be a fuzzy number on Rp with mðAÞ ¼ 0, and ni ðxÞ ¼ ai Aðx ci Þ, be a sequence of positive fuzzy numbers on Rp . P Suppose that a continuous Archimedean t-norm is given, limsupð ni¼1 an =nÞ 6 K, for some K 2 R; limðmax16i6n ai =nÞ ¼ 0, and mn ¼ ðc1 þ þ cn Þ=n ! casn ! 1. Then for any > 0,
SNðtÞ
P ¼ 0: limt!1 Cr
c
NðtÞ
By Proposition 3 and Lemma 2, the following is immediate from Theorem 8. 1 p ~i ðxÞg1 ~ Theorem 15. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm f is given, Ea ~1 < 1, and Ejj~c1 jj < 1. Then for that nn ðxÞðxÞ ¼ a any > 0,
SNðxÞðtÞ
P ¼ 0 a:s: ~ limt!1 Cr
E c 1
NðxÞðtÞ
Example 3. Let A be a fuzzy number on R2 defined by Aðx; yÞ ¼ maxf0; 1 ðx2 þ y2 Þg. Then ½A0 ¼ fðx; yÞ : x2 þ y2 6 1g. Let ~i ðxÞg1 fa i¼1 be a sequence of positive real valued i.i.d. random variables that are uniformly distributed on ½1; 3 and 2 f~ci ðxÞg1 i¼1 be a sequence of R -valued i.i.d. random variables that are uniformly distributed on ½3; 7 ½3; 11. Then ~1 < 1. Let fnn g be a sequence of fuzzy random variables such that nn ðxÞðxÞ ¼ a ~n ðxÞAðx ~cn ðxÞÞ. Then E~c1 ¼ ð5; 7Þ and Ea fnn g are positive fuzzy random variables and by Theorem 6, 12 and 13, we have, for any > 0,
Sn ðxÞ
< ¼ 1 a:s:; limn!1 Nes
ð5; 7Þ
n
NðxÞðtÞ 1 limn!1 Cr pffiffiffiffiffiffi P ¼ 0 a:s: t 74 and
limt!1
E½NðxÞðtÞ 1 ¼ pffiffiffiffiffiffi a:s: t 74
D.H. Hong / Information Sciences 228 (2013) 45–60
57
7. Results from the chance measure In this Section, we consider the law of large numbers, the renewal process for T-related fuzzy numbers with respect to the chance measure. Roughly speaking, a fuzzy random variable is a measurable function from a probability space to a collection of fuzzy numbers. A fuzzy random variable is a mapping n : X ! F ðRp Þ such that for any closed subset of C of Rp ,
n ðCÞðxÞ ¼ PosfnðxÞ 2 Cg ¼ suplnðxÞ ðxÞ x2C
is a measurable function of x, where lnðxÞ is the possibility distribution function of fuzzy variable nðxÞ. A scalar expected value is often required as a surrogate for a fuzzy random variable so that a decision-maker may employ the value to make a decision. For any fuzzy random variable n; nðxÞ is a fuzzy variable for all x 2 X. And E½nðxÞ, the expected value of nðxÞ, is a random variable defined on X. If we take the value E½nðxÞ as a surrogate for the n, then we have a novel definition of the expected value as follow. Definition 4 [18]. Let n be a fuzzy random variable. The expected value of n is defined as the expected value of E½nðxÞ, i.e.,
E½n ¼
Z Z X
1
Cr ðnðxÞ P r Þdr
Z
0
0
Cr ðnðxÞ 6 rÞdr dPðxÞ
ð11Þ
1
provided that at least one of the two integrals is finite. In particular, if n is a positive fuzzy random variable, then R R1 E½n ¼ X 0 Cr ðnðxÞ P r ÞdrdPðxÞ. Definition 5 [19]. Let n be a fuzzy random variable defined on the probability space ðX; A; PrÞ. The chance of an event n 2 B is defined as
Chfn 2 Bg ¼
Z
CrfnðxÞ 2 BgdPðxÞ:
X
It has been proved that (11) is equivalent to the following form (see [19,20]):
E½n ¼
Z 0
1
Chfn P rgdr
Z
0
Chfn 6 rgdr:
1
Definition 6 [30]. The sequence fnn g converges in chance to n if for any
> 0,
limn!1 Chfknn nk P g ¼ 0: A fuzzy random variable n is positive if and only if CrfnðxÞ 6 0g ¼ 0, for any x 2 X. By Lemma 2 and the bounded convergence theorem, the following is immediate from Theorem 9.
1 p ~i ðxÞg1 ~ Theorem 16. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm f is given, Ea ~1 < 1, and Ejj~c1 jj < 1. Then for that nn ðxÞðxÞ ¼ a any > 0,
Sn
~ P ¼ 0: limn!1 Ch
E c 1
n
By the bounded convergence theorem, the following is immediate from Theorem 15. 1 p ~i ðxÞg1 ~ Theorem 17. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm f is given, Ea ~1 < 1, and Ejj~c1 jj < 1. Then for that nn ðxÞðxÞ ¼ a any > 0,
SNðtÞ
~ P ¼ 0: limt!1 Ch
E c 1
NðtÞ 1 p ~i ðxÞg1 ~ Theorem 18. Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, fci ðxÞgi¼1 be a sequence of R -valued p i.i.d. random variables, A be a fuzzy number on R with mðAÞ ¼ 0, and fnn g be a sequence of positive fuzzy random variables such ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given. If Ea ~1 < 1 and Ejj~c1 jj < 1, then that nn ðxÞðxÞ ¼ a
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D.H. Hong / Information Sciences 228 (2013) 45–60
NðtÞ 1 P ¼ 0: limt!1 Ch t jjE~c1 jj Proof. According to Theorem 12, we know that fNðxÞðtÞ=tg converges in credibility to 1=jjE~c1 jj a.s. As a consequence, by bounded convergence theorem, we have
Z NðtÞ NðxÞðtÞ 1 1 P P ¼ lim dPðxÞ ¼ 0: limt!1 Ch Cr t!1 t Ejj~c1 jj t jjE~c1 jj X h ~i ðxÞg1 Lemma 5. Let A be a fuzzy variable on Rp with mðAÞ ¼ 0; fa i¼1 be a sequence of positive real valued i.i.d. random variables, 1 p ~ fci ðxÞgi¼1 be a sequence of R -valued i.i.d. random variables, and fnn g be a sequence of positive fuzzy random variables such that ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is given, 0 6 a1 ðxÞ 6 H a:s., and nn ðxÞðxÞ ¼ a jj~c1 ðxÞjj P M > 0 a:s. If M Hjj½A0 jj > 0, then for big r > 0,
NðtÞ P r ¼ 0: Ch t
Proof. Let I be an indicator function. As we have proved in Lemma 4,
Cr
NðxÞðtÞ Pr t
6 Pos
S½rt ðxÞ NðxÞðtÞ 1 P r ¼ sup ðzÞ : kzk 6 t r ½rt (P½rt
6 sup
i¼1 Iai ðxÞ½A0 þci ðxÞ ðzÞ
½rt
: kzk 6
1 r
)
8 9 < 1= P½rt P½rt 6 sup I ¼ supI ðzÞ : kzk 6 ðzÞ c ðxÞ : H½A þ i¼1 ci ðxÞ r; i¼1 i H½A þ \Bð0;1=rÞ 0
0
½rt
½rt
where ½rt is the smallest integer such that ½rt P rt. Then we have
inf fjjxjj : x 2 H½A0 þ
n X ci ðxÞ=ng P M Hjj½A0 jj > 0 a:s:; i¼1
for all n ¼ 1; 2; . . . and hence, for large r > 0, ½rt X c i ð xÞ
H½A0 þ
i¼1
½rt
\ Bð0; 1=rÞ ¼ ; a:s:
that is,
Cr
NðxÞðtÞ P r ¼ 0 a:s: t
Then we have, for large r > 0,
Z NðtÞ NðxÞðtÞ P r ¼ Cr P r dPðxÞ ¼ 0: Ch t t h ~i ðxÞg1 Theorem 19 (Fuzzy Renewal Theorem in Chance). Let fa i¼1 be a sequence of positive real valued i.i.d. random variables, 1 p ~ fci ðxÞgi¼1 be a sequence of R -valued i.i.d. random variables, A be a fuzzy number on Rp with mðAÞ ¼ 0, and fnn g be a sequence ~n ðxÞAðx ~cn ðxÞÞ. Suppose that a continuous Archimedean t-norm is of positive fuzzy random variables such that nn ðxÞðxÞ ¼ a given, 0 6 a1 ðxÞ 6 Ha:s., and jj~c1 ðxÞjj P M > 0a:s. If M Hjj½A0 jj > 0 and jjE~c1 jj < 1, then
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D.H. Hong / Information Sciences 228 (2013) 45–60
limt!1
E½NðtÞ 1 ¼ : t jjE~c1 jj
Proof. According to Theorem 12, we know that fNðxÞðtÞ=tg converges in credibility to 1=jjE~c1 jj. It follows from Liu [20, Theorem 3.54] that for any r 2 R,
limt!1 CrfNðxÞðtÞ=t P rg ¼ Cr
1 Pr jjE~c1 jj
a:s:;
and hence by Lebesgue dominated convergence theorem, we have
limt!1 ChfNðtÞ=t P rg ¼ limt!1
Z
CrfNðxÞðtÞ=t P rgdPðxÞ ¼ X
Z
Cr
X
1 1 P r dPðxÞ ¼ Ch Pr : jjE~c1 jj jjE~c1 jj
By Lemma 5, we note that, for large r > 0,
Ch
NðtÞ P r ¼ 0: t
As a consequence, by Lebesgue dominated convergence theorem, we have
limt!1
E½NðtÞ ¼ limt!1 t
Z
1
ChðNðtÞ=t P rÞdr ¼
0
1 : jjE~c1 jj
h Example 4. Let A be a fuzzy number on R2 defined by Aðx; yÞ ¼ maxf0; 1 ðx2 þ y2 Þg. Then ½A0 ¼ fðx; yÞ : x2 þ y2 6 1g. Let ~i ðxÞg1 fa i¼1 be a sequence of positive real valued i.i.d. random variables that are uniformly distributed on ½1; 3 and 2 f~ci ðxÞg1 i¼1 be a sequence of R -valued i.i.d. random variables that are uniformly distributed on ½3; 7 ½4; 10. Then ~ ~ Ec1 ¼ ð5; 7Þ; Ea1 ¼ 2; H ¼ 3; M ¼ 5, and jj½A0 jj ¼ 1. Let fnn g be a sequence of fuzzy random variables such that ~n ðxÞAðx ~cn ðxÞÞ. Then nn ðxÞðxÞ ¼ a
M Hjj½A0 jj ¼ 2 > 0: Then fnn g are positive fuzzy random variables and by Theorem 18, 19 and 20, we have, for any
SNðtÞ
P ¼ 0; limt!1 Ch
ð5; 7Þ
NðtÞ
> 0,
NðtÞ 1 pffiffiffiffiffiffi P ¼ 0 limn!1 Ch t 74 and
limt!1
E½NðtÞ 1 ¼ pffiffiffiffiffiffi : t 74
8. Conclusion This paper considers the law of large numbers and the renewal reward process in which the inter-arrival time is modeled as weighted fuzzy variables on Rp under continuous Archimedean t-norm-based fuzzy operations. The paper provides a far-reaching generalization of previous findings on the law of large numbers for T-related fuzzy numbers and derives different versions of the fuzzy renewal theorem for the rate of a renewal process having fuzzy inter-arrival times by using the law of large numbers for T-related weighted fuzzy variables. In addition, the paper considers the law of large numbers and the renewal process for Trelated fuzzy variables with respect to the credibility measure, the chance measure, and the expected value. Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0021089). References [1] R. Badard, The law of large numbers for fuzzy processes and the estimation problem, Information Science 28 (1982) 161–178. [2] G. Choquet, Theory of capacities Ann. Inst. Fourier, Grenoble 5 (1955) 131–295. [3] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets, World Scientific Publishing Co. Pte. Ltd., 1994.
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