Key Renewal Theory for T -iid Random Fuzzy Variables

Transcription

1 Applied Mathematical Sciences, Vol. 3, 29, no. 7, HIKARI Ltd, Key Renewal Theory for T -iid Random Fuzzy Variables Dug Hun Hong Department of Mathematics, Myongji University Yongin Kyunggido , South Korea This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 29 Hikari Ltd. Abstract In this paper, we investigate a Key renewal theory in the random fuzzy environment based on the concept of random fuzzy variable. As special cases, we consider the case for T min and T Archimedean t-norm. Mathematics Subject Classification: 6A86 Keywords: Random fuzzy process, infinite T -product possibility space, law of large numbers; renewal theories, Blackwell theory, Key renewal theory Introduction The theory of fuzzy sets, introduced by Zadeh 28, 29], has been widely examined and applied to statistics and the possibility theory in recent years. Generally, there are two approaches to deal with the phenomena combining randomness and fuzziness. One is applying the fuzzy random theory initiated by Kwakernaak 4, 5]. A fuzzy random variable is defined as a measurable function from probability space to the set of fuzzy variables. Since Kwakernaak 4, 5] introduced the concept of fuzzy random variables, there has been growing interest in fuzzy variables. A number of studies 6, 7, 8, 9,,, 2, 2, 22, 26, 27, 3, 32] have investigated renewal theory in the fuzzy random environment based on the concept of fuzzy variable and fuzzy random variable.

2 36 Dug Hun Hong The other is based on the random fuzzy theory presented by Liu 8]. This paper focuses on random fuzzy renewal processes. Liu 8] defined a new concept of random fuzzy variable as a function from a possibility space to the set of random variables. In order to rank random fuzzy variables, Liu and Liu 7] presented a scalar expected value operator. In addition, Liu 9] and Li and Liu 6] introduced the concept of independent and identically distributed random fuzzy variables. Zhu and Liu 32] presented the concept of chance distribution to describe random fuzzy variable. In particular, Zhao et al. 25] built a random fuzzy renewal process model with a random fuzzy renewal theorem and established the respective version of Blackwell s theorem. Shen et al 25] presented the result of an investigation into the representation of properties of alternating renewal process that are described by sequences of positive random vectors. It is noted that all of these studies on random fuzzy processes used /min-norm based fuzzy operations. In general, we can consider the extension principle realized by the means of some t-norm T. In this paper, we construct an T -independent and identically distributed random fuzzy process on infinite T -product possibility space. We investigate law of large numbers, renewal theories, Blackwell theory and Key renewal theory in the random fuzzy environment based on the concept of fuzzy variable and random fuzzy variable. As special cases, the case for T min and T Archimedean t-norm are treated. 2 Preliminaries We begin by reviewing some concepts and results concerning fuzzy variables. Let ξ be a fuzzy variable with a possibility distribution (membership function) µ on a possibility space (Θ, P(Θ), P os), where Θ is a universe, P(Θ) is the power set of Θ and P os is a possibility measure defined on P(Θ). For a fuzzy variable ξ and any subset D of the real numbers R, the quantity Nes{ξ D} : sup µ ξ (x) : P os{ξ D c } x/ D is considered to measure the necessity of ξ belonging to D (see 2]). The credibility of ξ belonging to D and the expected value Eξ] (]) are defined as Cr{ξ D} (P os{ξ D} + Nes{ξ D}), 2 Eξ] Cr{ξ r}dr Cr{ξ r}dr provided that at least one of the two integrals is finite. In particular, if ξ is a nonnegative fuzzy variable (i.e., Cr{ξ < } ), then Eξ] Cr{ξ r}dr.

3 Key renewal theory for T -iid random fuzzy variables 37 Let ξ be a fuzzy variable on a possibility space (Θ, P(Θ), P os). Then its membership function µ ξ is determined from the credibility measure by µ ξ (x) (2Cr{ξ x}), x R. Let ξ be a fuzzy variable a the possibility space (Θ, P(Θ), P os). Then, for (, ] and for ξ inf{x µ ξ (x) } and ξ sup{x µ ξ (x) } ξ inf {x µ ξ (x) > } and ξ sup {x µ ξ (x) > } are called the -pessimistic value and the -optimistic value of ξ, respectively. If ξ is a fuzzy variable with finite expected value Eξ], then Eξ] 2 ξ + ξ ]d. Let ξ] be the -level sets with ξ] {x R µ ξ } for (, ], and ξ cl{x R µ ξ > }. It is noted that if µ ξ is fuzzy convex and upper semi-continuous function, then ξ] ξ, ξ ]. Let K(R) denote the class of nonempty compact convex subsets of R The linear structure induced by the scalar product and the Minkowski addition is that λa {λa a A}, A + B {a + b a A, b B}, for all A, B K(R), and λ R. If d H is the Hausdorff metric on K(R), which for A, B K(R) is given by d H (A, B) max { sup inf a A b B a b, sup inf a b b B a A then (K(R), d H ) is a complete and separable metric space ]. We note that if A a, a 2 ], B b, b 2 ], then d H (A, B) max { a b, a 2 b 2 }. The norm of an element of K(R) is denoted by A d H (A, {}) sup{ x : x A}. Recall that a triangular norm (or a t-norm) is a commutative monoid operation in, ] with neutral element and is monotonic (non-decreasing) when viewed as a bivariate function. A t-norm T is said to be Archimedean if T (x, x) < x for all x (, ). It is easy to check that the minimum t-norm },

4 38 Dug Hun Hong is not Archimedean. From representation theorem in topological semigroup theory 7], every continuous t-norm T is uniquely representable as an ordinal sum where in each summand the corresponding t-norm is Archimedean. This means that there is a finite or countable index set I and a family of subintervals {a i, b i ]} i I for which a i b i, having non-overlapping interiors and covering, ], such that the following holds: Let φ i : a i, b i ], ] be the natural homomorphism, that is, φ i (a) a a i b i a i, a a i, b i ]. And φ i : a i, b i ] a i, b i ], ], ] is defined by φ i (a, b) (φ i (a), φ i (b)), a, b a i, b i ]. (i) There exists a subset I I such that for i I, the restriction of T to a i, b i ] a i, b i ] is T i φ i, that is, T ai,b i ] a i,b i ](x, y) T i φ i (x, y). where T i is an Archimedean t-norm. (ii) Elsewhere, T is the minimum. We note that T (x, y) min(x, y) if (x, y) / We shall write T (< a i, b i, T i >) i I. We define K : F(R) F(R) by i I ((a i, b i ) (a i, b i )). { u]bi if (a Ku] i, b i ] for i I, u] otherwise. The fuzzy variables ξ on the possibility space (Θ, P(Θ ), P os ) and ξ 2 on the possibility space (Θ 2, P(Θ 2 ), P os 2 ) are said to be identically distributed if and only if P os{ξ B} P os{ξ 2 B} for any sets B of R. Let T be a t-norm. A family of fuzzy variables {ξ i, i I} is called T - independent if for any subset {i, i 2,, i n } I with n 2, P os{ξ ik B k, k, 2,, n} T n kp os{ξ ik B k }, for any subsets B, B 2,, B n of R. For T -independent fuzzy variables ξ k, k m with possibility distributions µ k, k m, and a function g : R m R, the possibility distribution of

5 Key renewal theory for T -iid random fuzzy variables 39 g(ξ, ξ 2,, ξ m ) is determined via the possibility distributions µ ξ, µ ξ2,, µ ξm as µ g(ξ,ξ 2,,ξ m)(x) Pos{g(ξ, ξ 2,, ξ m ) x} sup Tkµ m ξk (x k ), x,x 2,,x m R, xg(x,x 2,,x m) where T can be any general t-norm. This is the (generalized) extension principle associated with t-norm. A fuzzy number ξ is a fuzzy variable of the real line with a normal ( there exist x R such that µ ξ (x) ), fuzzy convex and upper semi-continuous membership function and ξ] is bounded for each (, ]. A random fuzzy variable 2] is a function from a possibility space (Θ, P(Θ), P os) to a collection of random variables F. The expected value of random fuzzy variable is defined by Liu ] as Eξ] Cr{θ Θ Eξ(θ)] r}dr Cr{θ Θ Eξ(θ)] r}dr. Definition 2. Random fuzzy variables ξ, ξ 2,, ξ n are said to be T - independent if (a) ξ (θ), ξ 2 (θ),, ξ n (θ) are independent random variables for each θ; (b) Eξ ( )], Eξ 2 ( )],, Eξ n ( )] are T -independent fuzzy variables. It is noted that for a random fuzzy variables ξ and a Borel set B of R, P {ξ( ) B} is a fuzzy variable. Definition 2.2 The random fuzzy variables ξ and η are said to be identically distributed if for any element B of Borel field B of R, P {ξ( ) B} and P {η( ) B} are identically distributed fuzzy variables. We briefly review a construction of a sequence of T -iid random fuzzy variables. Let (Θ, P(Θ), P os) be a possibility space and F be a family of distributions of random variables. Let ξ : Θ F be a random fuzzy variable. We denote by Θ Π iθ the space consisting of all infinite sequences of probability distribution functions (θ, θ 2, ), θ n Θ and R Π ir the space consisting of all infinite sequences (x, x 2, ) of real numbers. We take B to be the Borel σ-field of R. Define a possibility measure P os P os on Θ such that for any A Θ, P os {A} sup T (P os{θ }, P os{θ 2 },, ). (θ,θ 2, ) A

6 32 Dug Hun Hong Then (Θ, P(Θ ), P os ) is called the T -product possibility measure of (θ, θ 2, ). Let P θi be the probability measure on R with probability distribution θ i. For each θ (θ, θ 2, ) define a probability measure on (R, B ) so that P θ Π ip θi, the product probability measure of P θi, i, 2,. Define a process {X n } on (R, B ) such that X n (x, x 2, ) x n. By the definition of P θ, the process {X n } is independent with respect to P θ and θ n is the probability distribution of X n. We now define a random fuzzy variables {ξ n } on (Θ, P(Θ ), P os ) such that ξ n ( θ) X n with respect to P θ and set S, S n ξ + ξ ξ n, n, 2,. Then, by Theorem 2 6], the random fuzzy variables ξ n, n, 2, on (Θ, P(Θ ), P os ) are T -iid random fuzzy variables and identically distributed with a random fuzzy variable ξ. 3 Random fuzzy Key renewal theory From this section, we assume that {ξ n } on (Θ, P(Θ ), P os ) is a T - independent and identically distributed random fuzzy process defined in section 4 and t Eξ ( θ)] is a fuzzy number. In this section, we shall discuss the renewal theory of random fuzzy process. From this section, we additionally assume that Θ is a set of probability distribution functions such that θ(), θ() <. Let {ξ n } be a T -independent and identically distributed random fuzzy process on (Θ, P(Θ ), P os ). Let ξ n denote the times between the (n )th and the nth events, known as the inter-arrival times, n, 2,, respectively. Define S, S n ξ + ξ ξ n, n, If the inter-arrival times ξ n, n, 2, are random fuzzy variables then the process {S n, n } is called a random fuzzy renewal process. Let N(t) denote the total number of the events that have occurred by time t. Then we have N(t) max{n < S n t}. For any fixed θ (θ, θ 2, ) Θ, it is clear that N(t)( θ) is a random variable with the probability distribution P {N(t)( θ) n} P {S n ( θ) t} P {S n+ ( θ) t}, n, 2,, where S n ( θ) n i ξ i ( θ) n i X i w.r.t. P θ. We call N(t) the random fuzzy renewal variable. For each θ Θ, EN(t)( θ) is the expected values of the random variables N(t)( θ). However, when θ is varied all over in Θ, EN(t)( θ)], as function of θ Θ, is fuzzy variable and their -pessimistic and -optimistic values can

7 Key renewal theory for T -iid random fuzzy variables 32 be expressed by EN(t)( θ)] inf{t µ EN(t)( θ)] (t) }, EN(t)( θ)] sup{t µ EN(t)( θ)] (t) }. To begin with, we recall some important results for renewal process of classical stochastic process. Let {X n, n, 2, } be a sequence of independent nonnegative random variables with X having distribution G, and X n having distribution F, n >. Let U n n X i, n and define N D (t) max{n : < U n t}. The stochastic process {N D (t), t > } is called a delayed renewal process. A sufficient condition for h to be directly Riemann integrable is that h is a positive nonincreasing function of t on, ) such that <. Proposition 3. (23]) Let µ xdf (x) and m D (t) EN D (t)]. ) If F is not lattice, then lim EN D(t + a)] EN D (t)] a µ. 2) If F is not lattice, µ <, and h directly Riemann integrable, then lim h(t x)dm D (x). µ Let {X n, n, 2, } be a sequence of independent nonnegative random variables with X i having distribution G i for i, 2,, n, and X n having distribution F, n > n. Let Y n i X i and Y n X n +(n ), n >. Let V n n X i, V n n Y i, n and define Then we have N(t) max{n : < V n t}, N D(t) max{n : < V n t}. N D(t) + (n ) N(t). From this factor, we have the following lemma by Proposition 3.. Proposition 3.2 Let µ xdf (x) and m(t) EN(t)]. ) If F and G i, i, 2,, n are not lattice, then lim EN(t + a)] EN(t)] a µ.

8 322 Dug Hun Hong 2) If F is not lattice, µ <, and h directly Riemann integrable, then lim h(t x)dm(x). µ Definition 3.3 The set Θ is said to be a totally ordered set with the stochastic ordering, if for any probability distribution functions θ, θ in Θ such that θ θ and r R, either θ(r) θ (r) denoted by θ d θ, or θ(r) θ (r) denoted by θ d θ. Lemma 3.4 Suppose Θ is totally ordered set with the stochastic ordering. Let / i I (a i, b i ). Then for any (, ], we have A { θ (θ, θ 2, ) Θ P os ( θ) } { θ (θ, θ 2, ) Θ P os(θ i ), i, 2, } { θ (θ, θ 2, ) Θ θ d θ i d θ, i, 2, }. Lemma 3.5 Suppose Θ is totally ordered set with the stochastic ordering. Let A { θ (θ, θ 2, ) Θ P os ( θ) } and let (a j, b j ), j I. Then for any δ > with b j δ >, there exist M δ > and θ b j δ (θ b j δ,, θ b j δ,2, ), θ b j δ (θ b j δ,, θ b j δ,2, ) such that θ b j δ,k θ b j δ, θ b j δ,k θ b j δ for k > M δ and EN(t)( θ b j δ)] EN(t)(θ)], EN(t)(θ)] EN(t)( θ b j δ)]. In this paper, we assume that h is a positive nonincreasing function of t on, ) such that <. Theorem 3.6 Let T be a continuous t-norm and let Θ be a totally ordered set with the stochastic ordering. Let {ξ n } be a T -independent and identically distributed random fuzzy process on (Θ, P(Θ ), P os ) such that Eξ ( θ)] <, (, ]. If any θ Θ is nonlattice probability distribution function, then we have, for (, ], lim d H ( t h(t x)den(t)( θ)], KEξ ( θ)] ] ). Proof. Since µ Eξ ( θ)] (t) is fuzzy convex and upper semi continuous and Θ is a totally ordered set with the stochastic ordering, for (, ] there exist θ, θ Θ such that {θ Θ : P os(θ) } {θ Θ : µ Eξ ( θ))] (t) } {θ Θ : θ d θ d θ }.

9 Key renewal theory for T -iid random fuzzy variables 323 We note that { θ Θ : µ EN(t)( θ))] (t) } { θ (θ, θ 2, ) Θ : P os ( θ)) } A. There are two possible cases. We first consider the case for / i I (a i, b i ). Let θ (θ, θ, ), θ (θ, θ, ) Θ. Then, since Θ is a totally ordered set with the stochastic ordering, we clearly have by Lemma 3.4, h(t x)den(t)( θ)] h(t x)den(t)( θ)] Then, by Proposition 3., we have and h(t x)dm θ (x), h(t x)dm θ (x). lim h(t x)den(t)( θ)] lim h(t x)dm θ (x) Eξ ( θ )] ] () KEξ ( θ)] lim h(t x)den(t)( θ)] lim h(t x)dm θ (x) Eξ ( θ )] ] (2) KEξ ( θ)] Hence, the result follows. We now consider (a j, b j ), j I. By Lemma 3.5, we have for any δ > with b j δ >, there exist θ b j δ (θ b j δ,, θ b j δ,2, ), θ b j δ (θ b j δ,, θ b j δ,2, ) Θ such that h(t x)dm θ (x) h(t x)den(t)( θ)] b j δ, h(t x)dm θ (x) bj h(t x)den(t)( θ)] δ, Then we have, by Proposition 3.2, lim inf h(t x)den(t)( θ)] lim h(t (x) x)dm θ b j δ

10 324 Dug Hun Hong xdθ b j δ (x), and lim sup h(t x)den(t)( θ)] lim h(t x)dm θ (x) bj δ xdθ bj δ (x), where the two equalities come from Proposition 3.2. Since δ > is arbitrary and µ Eξ (θ)](t) is upper semi-continuous, by letting δ, we have, by Proposition 3.2. and lim inf h(t x)den(t)( θ)] lim sup h(t x)den(t)( θ)] On the other hand, since and we have lim sup h(t x)den(t)( θ)] h(t x)den(t)( θ)] b j xdθ b j (x) Eξ (θ)]] KEξ ( θ)] b j ], (3) xdθ bj (x) Eξ (θ)]] b j ]. (4) KEξ ( θ)] h(t x)den(t)( θ)] b j, h(t x)den(t)( θ)], h(t x)den(t)( θ)] lim h(t x)den(t)( θ)] b j Eξ (θ)]] b j KEξ ( θ)] ], (5)

11 Key renewal theory for T -iid random fuzzy variables 325 and, lim inf h(t x)den(t)( θ)] lim h(t x)den(t)( θ)] b j Eξ (θ)]] b j ]. (6) KEξ ( θ)] Therefore, we have from (3) and (5) t lim h(t x)den(t)( θ)] KEξ ( θ)] from (4) and (6) t lim h(t x)den(t)( θ)] KEξ ( θ)] which proves the result. ] ], (7), (8) Corollary 3.7 Let T be an Archimedean continuous t-norm and let Θ be a totally ordered set with the stochastic ordering. Let {ξ n } be a T -independent and identically distributed random fuzzy process on (Θ, P(Θ ), P os ) such that Eξ ( θ)] <, (, ]. If any θ Θ is nonlattice probability distribution function, then we have, for (, ], lim d H ( ) h(t x)den(t)( θ)], Eξ ( θ)], Corollary 3.8 Let T min and let Θ be a totally ordered set with the stochastic ordering. Let {ξ n } be a T -independent and identically distributed random fuzzy process on (Θ, P(Θ ), P os ) such that Eξ ( θ)] <, (, ]. If any θ Θ is nonlattice probability distribution function, then we have, for (, ], lim d H ( ) h(t x)den(t)( θ)], Eξ ( θ)], It is noted that a class of scale densities is a totally ordered set with the stochastic ordering. In the next result, we assume that µ Eξ ( θ)] (t) is a fuzzy number, and consider classical version of Key renewal theory for T -iid random fuzzy variables.

12 326 Dug Hun Hong Example 3.9 Let T (<, 2/3, T >, < 2/3,, T 2 >) and Θ {θ σ < σ < } be a class of scale densities of exponential distribution with mean parameter σ. Let Eξ ( θ)], Eξ ( θ)] 2 then We also have Eξ ( θ)] 2, Eξ ( θ)]. { 2 KEξ ( θ)]] for, 2/3], 3 for (2/3, ], { 4 KEξ ( θ)]] for, 2/3], 3 for (2/3, ]. and hence ] KEξ ( θ)] KEξ ( θ)]] ] KEξ ( θ)] KEξ ( θ)]] { 3 4 for, 2/3], for (2/3, ], { 3 2 for, 2/3], for (2/3, ]. If h(x) /x 2, x, ) and h(x), x (, ), then h(x)dx 2, then by Theorem 3.6, lim d H ( t h(t x)den(t)( θ)], 2 KEξ ( θ)] ] ). Acknowledgements. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (27RDAB27869). References ] P. Billingsley, Probability and Measure, John Wiley and Sons, Inc ] K. L. Chung, A Course in Probability Theory, Second Edition, Academic Press, Inc ] P. Diamond, P. Kloeden, Metric space of fuzzy sets, World Scientific Publishing Co. Pte. Ltd., ] D. Dubois, H. Prade, Fuzzy Sets and Systems, Mathematics in science and Engineering, Inc. 978.

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15 Key renewal theory for T -iid random fuzzy variables ] R. Zhao, and Tang, W. Some properties of fuzzy random processes, IEEE Transactions on Fuzzy Systems, 2(26), ] R. Zhao, W. Tang, H. Yun, Random fuzzy renewal process, European Journal of Operational Research, 69(26), ] R. Zhao, W. Tang and C. Wang, Fuzzy random renewal process and renewal reward process, Fuzzy Optimization and Decision Making, 6(27), ] Y. Zhu, B. Liu. Continuity theorems and chance distribution of random fuzzy variable, Proceedings of the Royal Society of London Series A, 46(24), Received: March 5, 29; Published: March 22, 29