A Note on Renewal Theory for T -iid Random Fuzzy Variables

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1 Applied Mahemaical Sciences, Vol, 6, no 6, HIKARI Ld, wwwm-hikaricom hp://dxdoiorg/988/ams6686 A Noe on Renewal Theory for T -iid Rom Fuzzy Variables Dug Hun Hong Deparmen of Mahemaics, Myongji Universiy Yongin Kyunggido , Souh Korea Copyrigh c 6 Dug Hun Hong This aricle is disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, reproducion in any medium, provided he original work is properly cied Absrac In his noe, we invesigae a classical version of renewal heories in he T -independen idenically disribued rom fuzzy variables For special cases, we consider he case for T min T Archimedean -norm Mahemaics Subjec Classificaion: 6A86 Keywords: T -iid rom fuzzy variables; he law of large numbers; renewal heories Inroducion preliminaries A number of sudies -9] have invesigaed renewal heory in he rom fuzzy environmen based on he concep of fuzzy variable rom fuzzy variable Recenly, Hong ] invesigaed renewal heories in he T -independen rom fuzzy environmen based on he concep of rom fuzzy variable including he cases for T min T Archimedean -norm In his noe, we consider a classical version of renewal heories in he T -iid rom fuzzy variables For special cases, we consider he case for T min T Archimedean -norm For basic noaions definiions for fuzzy variables heir T -norm based operaions, please refer o he paper ] A rom fuzzy variable 6] is a funcion from a possibiliy space (Θ, P(Θ, P os o a collecion of rom variables F The expeced value of rom fuzzy

2 97 Dug Hun Hong variable is defined by Liu Liu 4] as Eξ] Cr{θ Θ Eξ(θ] r}dr Cr{θ Θ Eξ(θ] r}dr Definiion ] Rom fuzzy variables ξ, ξ,, ξ n are said o be T - independen if (a ξ (θ, ξ (θ,, ξ n (θ are independen rom variables for each θ; (b Eξ ( ], Eξ ( ],, Eξ n ( ] are T -independen fuzzy variables I is noed ha for a rom fuzzy variables ξ a Borel se B of R, P {ξ( B} is a fuzzy variable Definiion ] The rom fuzzy variables ξ η are said o be idenically disribued if for any elemen B of Borel field B of R, P {ξ( B} P {η( B} are idenically disribued fuzzy variables Le Θ be a family of probabiliy disribuion funcions on R le (Θ, P(Θ, P os be a possibiliy space F be a family of disribuions of rom variables Le ξ : Θ F be a rom fuzzy variable We denoe by Θ Π iθ he space consising of all infinie sequences of probabiliy disribuion funcions (θ, θ,, θ n Θ R Π ir he space consising of all infinie sequences (x, x, of real numbers We ake B o be he Borel σ-field of R Define a possibiliy measure P os on Θ such ha for any A Θ, P os {A} sup T (P os{θ }, P os{θ },, (θ,θ, A Then (Θ, P(Θ, P os is called he T -produc possibiliy measure of (θ, θ, Le P θi be he probabiliy measure on R wih probabiliy disribuion θ i For each θ (θ, θ, define a probabiliy measure on (R, B so ha P θ Π ip θi, he produc probabiliy measure of P θi, i,, Define a process {X n } on (R, B such ha X n (x, x, x n By he definiion of P θ, he process {X n } is independen wih respec o P θ θ n is he probabiliy disribuion of X n We now define a rom fuzzy variables {ξ n } on (Θ, P(Θ, P os such ha ξ n ( θ X n wih respec o P θ se S, S n ξ + ξ + + ξ n, n,, Then, by Theorem ], he rom fuzzy variables ξ n, n,, on (R, B are T -iid rom fuzzy variables idenically disribued wih a rom fuzzy variable ξ Rom fuzzy renewal heories From his secion, we addiionally assume ha Θ is a se of probabiliy disribuion funcions such ha θ(, θ( < Le ξ n denoes he ime beween

3 Renewal heory for T -iid rom fuzzy variables 97 he (n h he nh evens, known as he iner-arrival imes, n,,, respecively Define S, S n ξ + ξ + + ξ n, n, If he iner-arrival imes ξ n, n,, are rom fuzzy variables hen he process {S n, n } is called a rom fuzzy renewal process Le N( denoes he oal number of he evens ha have occurred by ime Then we have N( max{n < S n } For any fixed θ (θ, θ, Θ, i is clear ha N(( θ is a rom variable wih he probabiliy disribuion P {N(( θ n} P {S n ( θ } P {S n+ ( θ }, n,,, where S n ( θ n i ξ i( θ n i X i wr P θ We call N( he rom fuzzy renewal variable For each θ Θ, EN(( θ is he expeced values of he rom variables N(( θ However, when θ is varied all over in Θ, EN(( θ], as a funcion of θ Θ, is fuzzy variable heir -pessimisic -opimisic values can be expressed by EN(( θ] inf{ µ EN(( θ] ( }, EN(( θ] sup{ µ EN(( θ] ( } Recenly, Hong ] invesigaed rom fuzzy elemenary renewal heories for T -iid rom fuzzy variables as follows Theorem ] Le {ξ n } be a T -iid rom fuzzy process on (Θ, P(Θ, P os such ha Eξ ( θ] <, (, ] Then we have, for (, ], ( ] EN(( θ] d H, as KEξ ( θ] Corollary ] Le {ξ n } be a T -iid rom fuzzy process on (Θ, P(Θ, P os Suppose ha Eξ ( θ] <, (, ] T is an Archimedean -norm, hen we have, for all < ( EN(( θ] d H, Eξ ( θ] as Corollary ] Le {ξ n } be a T -iid rom fuzzy process on (Θ, P(Θ, P os Suppose ha Eξ ( θ] <, (, ] T min, hen we have, for all < ( ] EN(( θ] d H, Eξ ( θ] as

4 974 Dug Hun Hong A scale densiy is a densiy of he form ( x σ f σ where σ > The parameer σ is called a scale parameer The following lemma is easy o check Lemma Le f be a densiy funcion wih E f xf(xdx < Le θ σ (x x σ f ( y σ dy, hen θ σ (σa θ (a xdθ σ (x σ xf(xdx Example The followings are examples of scale densiies Normal densiy N(, σ : f(x σ Gamma densiy Γ(, β ( fixed: f(x, β Uniform densiy U(, β: /σ (π / e x, σ Γ(β y e y/β I (, (x, f(x β β I (,β(x -disribuion wih degree of freedom T (, σ ( > fixed: (+/ f(x, σ Γ( + /] ( + x σ(π / σ I is noed ha a class of scale densiies is a oally ordered se wih he sochasic ordering In he nex resul, we assume ha Θ is a class of scale densiies µ Eξ ( θ] ( is a fuzzy number, consider classical version of renewal heories for T -iid rom fuzzy variables Theorem Le Θ {θ σ < σ < } be a class of scale densiies of f Le {ξ n } be a T -iid rom fuzzy process on (Θ, P(Θ, P os If E Eξ <, hen we have ( θ] ] EN(] lim E KEξ ( θ]

5 Renewal heory for T -iid rom fuzzy variables 975 We need he following lemma Lemma Le Θ {θ σ < σ < } be a class of scale densiies of f Le {ξ n } be a T -iid rom fuzzy process on (Θ, P(Θ, P os Then, we have sup > EN(( θ ] C Eξ ( θ] for some consan C > Proof Since µ Eξ ( θ] ( is fuzzy convex upper semi coninuous Θ is a oally ordered se wih he sochasic ordering, for (, ] here exis θ, θ Θ such ha {θ Θ : P os(θ } {θ Θ : µ Eξ ( θ] ( } {θ Θ : θ d θ d θ } Then we clearly have EN(( θ ] EN(( θ]] EN(( θ]] EN(( θ ] where θ (θ, θ, θ (θ, θ, Le θ θ f(, θ θ f( Then f ( is bounded increasing funcion f ( is decreasing funcion such ha f ( f (, since µ Eξ ( θ] ( is fuzzy convex We also noe ha by Lemma, Eξ ( θ] xdθ f( (x f (E f We chose a > such ha < θ (af ( < le p θ (af ( Define new iner-arrival imes via runcaion ˆX n af (I{Xn > af (} Thus ˆX n wih probabiliy θ (af ( equals af ( wih probabiliy θ (af ( Le ˆN(( θ denoe he couning process obained by using hese new iner-arrival imes, i follows ha N(( θ ˆN(( θ, > Leing H n ( θ denoe he number of arrivals ha occurs a ime naf (, we conclude ha {H n ( θ } is iid wih a geomeric disribuion wih success probabiliy p Leing x] denoe he smalles ineger x, we have he inequaliy N(( θ ˆN(( θ H(( θ Observing ha p p by Lemma, /af (] n H n ( θ, > E(H(( θ /af i (]E(H ( θ ( af ( + + af (, p af (p

6 976 Dug Hun Hong we obain EN(( θ ] ( + af ( af ( p Since < f ( f (, here exis consans C > such ha for >, + af ( af ( + af ( af ( af ( + f ( f ( C f (, hence EN(( θ ] Proof of Theorem We firs noe ha EN(] ( EN(( θ] ] E KEξ ( θ] C E f p Eξ ( θ] ( KEξ ( θ] + EN(( θ] d ] + KEξ ( θ] From Theorem, we have ha, for (, ], EN(( θ] lim KEξ ( θ] lim EN(( θ] Hence, i suffices o prove ha lim lim EN(( θ] d EN(( θ] d KEξ ( θ] ] ] KEξ ( θ] KEξ ( θ] By Lemma, we have for >, ( EN(( θ] ( EN(( θ] d d ( EN(( θ ] d C ] ] ] d d d ( EN(( θ ] d CE Eξ ( θ] Eξ ( θ] d

7 Renewal heory for T -iid rom fuzzy variables 977 Therefore, by Dominaed Convergence Theorem, we immediaely have he resul If T min, we have he following resul Corollary Le T min Θ {θ σ < σ < } be a class of scale densiies of f Le {ξ n } be a T -iid rom fuzzy process on (Θ, P(Θ, P os If E Eξ <, hen we have ( θ] ] EN(] lim E Eξ ( θ] If T is an Archimedean -norm, we have he following resul Corollary 4 Le T be an Archimedean -norm Θ {θ σ < σ < } be a class of scale densiies of f Le {ξ n } be a T -iid rom fuzzy process on (Θ, P(Θ, P os If E Eξ <, hen we have ( θ] EN(] lim ( + Eξ ( θ] Eξ ( θ] Example Le T (<, /, T >, < /,, T > Θ {θ σ < σ < } be a class of scale densiies of exponenial disribuion wih mean parameer σ wih Then Then We also have µ Eξ ( θ](σ { σ for σ, ], σ for σ, ], oherwise Eξ ( θ], Eξ ( θ] Eξ ( θ] µ (σ Eξ ( θ], Eξ ( θ] { σ for σ /, ], σ for σ,, σ oherwise { KEξ ( θ]] for, /], for (/, ], { 4 KEξ ( θ]] for, /], for (/, ]

8 978 Dug Hun Hong hence Then We have similarly, KEξ ( θ] KEξ ( θ] ] ] { for, /], KEξ ( θ]] 4 for (/, ], { for, /], KEξ ( θ]] for (/, ] µ (σ KEξ ( θ] ] E Eξ ( θ] { for σ, for σ /4, (, /], oherwise ( + d ( + log ] E KEξ ( θ] ( If T be an coninuous Archimedean -norm, hen by Theorem ( EN(] lim If T min, hen by Corollary lim EN(] ( + log If T be an Archimedean -norm, hen by Corollary 4 lim EN(] References ] D H Hong, Renewal heory for T -iid rom fuzzy variables, Applied Mahemaical Sciences, (6, hp://dxdoiorg/988/ams66575 ] D H Hong, J D Kim, The exisence of T -iid rom fuzzy variables is law of large numbers, Inernaional Journal of Mahemaical Analysis, 9 (5, hp://dxdoiorg/988/ijma5589

9 Renewal heory for T -iid rom fuzzy variables 979 ] X Li, B Liu, New independence definiion of fuzzy rom variable rom fuzzy variable, World Journal of Modeling Simulaion, (6, 8-4 4] Y K Liu, B Liu, Expeced value operaor of rom fuzzy variable rom fuzzy expeced value models, Inernaional Journal of Uncerainy, Fuzziness Knowledge-Based Sysems, (, 95-5 hp://dxdoiorg/4/s ] B Liu, Theory Pracice of Uncerain Programming, Physica-Verlag, Heidelberg, 6] B Liu, Uncerainy Theory: An Inroducion o is Axiomaic Foundaions, Springer-Verlag, Berlin, 4 hp://dxdoiorg/7/ ] Q Shen, R Zhao, W Tang, Rom fuzzy alernaing renewal process, Sof Compuing, (9, 9-47 hp://dxdoiorg/7/s5-8-7-y 8] R Zhao, W Tang C Wang, Fuzzy rom renewal process renewal reward process, Fuzzy Opimizaion Decision Making, 6 (7, hp://dxdoiorg/7/s7-7-9-z 9] Y Zhu, B Liu, Coninuiy heorems chance disribuion of rom fuzzy variable, Proceedings of he Royal Sociey of London Series A: Mahemaical, Physical Engineering Sciences, 46 (4, hp://dxdoiorg/98/rspa48 Received: Augus 9, 6; Published: Ocober 6, 6

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