Fuzzy random variables and Kolomogrov s important results

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1 Iteratioal Joural of Egieerig Sciece Ivetio ISSN (Olie): , ISSN (Prit): Volume 5 Issue 3 March 06 PP.45-6 Fuzzy radom variables ad Kolomogrov s importat results Dr.Earest Lazarus Piriya kumar, R. Deepa, Associate Professor, Departmet of Mathematics, Traquebar Bishop Maickam Luthera College, Porayar, Idia. Assistat Professor, Departmet of Mathematics, E.G.S. Pillay Egieerig College, Nagapattiam, Tamiladu, Idia. ABSTRACT :I this paper a attempt is made to trasform Kolomogrov Maximal iequality, Koroecker Lemma, Loeve s Lemma ad Kolomogrov s strog law of large umbers for idepedet, idetically distributive fuzzy Radom variables. The applicatios of this results is extesive ad could produce itesive isights o Fuzzy Radom variables. Keywords :Fuzzy Radom Variables, Fuzzy Real Number, Fuzzy distributio fuctio, Strog law of Large Numbers. I. Itroductio The theory of fuzzy radom variables ad fuzzy stochastic processes has received much attetio i recet years [-]. Prompted for studyig law of large umbers for fuzzy radom variables is both theoretical sice of major cocer i fuzzy stochastic theory as i the case of classical probability theory would be the differet limit theorems for sequeces of fuzzy radom variables ad practically sice they are applicable to statistical aalysis whe samples or prior iformatio are fuzzy. The cocept of fuzzy radom variables was itroduced by Kwakerack [4] ad Puri ad Ralesea [6]. I order to make fuzzy radom variables applicable to statistical aalysis for imprecise data, we eed to come up with weak law of large umbers, strog law of large umbers ad Kolomogorov iequalities. I the preset paper we have deduced Kolomogrov maximal iequality, Kroecker s lemma, ad Loeve s Lemma.. Prelimiaries I this sectio, we describe some basic cocepts of fuzzy umbers. Let R deote the real lie. A fuzzy umber is a fuzzy set u : R [0, ] with the followig properties. ) u is ormal, i.e. there exists x R such that u (x). ) u is upper semicotiuous. 3) Suppu cl{x R u x > 0} is compact. rights reserved. 4) u is a covex fuzzy set, i.e. u λx λ y mi (u(x)), u (y)) for x, y, R ad λ [0, ]. Let F(R) be the family of all fuzzy umbers. For a fuzzy set u, if we defie. L a u x: u x, 0 <, Supp u 0 The it follows that u is a fuzzy umber if ad oly if L u ad L a u is a closed bouded iterval for each λ [0, ]. From this characterizatio of fuzzy umbers, a fuzzy umber u is a completely determied by the ed poits of the itervals L a u [u x,u x ]. 3. Fuzzy Radom Variables: Throughout this paper, (Ω, A, P) deotes a complete probability space. 45 Page

2 Fuzzy radom variables ad Kolomogrov s If x : Ω F(R) is a fuzzy umber valued fuctio ad B is a subset of R, the X (B) deotes the fuzzy subset of Ω defied by X B ω Sup x B X(ω)(x) for every ω Ω. The fuctio X : Ω F(R) is called a fuzzy radom variable if for every closed subset B or R, the fuzzy set X B is measurable whe cosider as a fuctio from Ω to [0,]. If we deote X(ω) { X x (ω), X x (ω) 0, the it is a well-kow that Xis a fuzzy radom variable if ad oly if for each [0, ]. X x ad X x are radom variable i the usual sese (for details, see Ref.[]). Hece, if σ X is the smallest σ field which makes Xis a cosistet with σ({x x,x x 0 }). This eables us to defie the cocept of idepedece of fuzzy radom variables as i the case of classical radom variables. 4. Fuzzy Radom Variable ad its Distributio Fuctio ad Exceptio Give a real umber, x, we ca iduce a fuzzy umber u with membership fuctio ξ x (r) such that ξ x (x) < for r x (i.e. the membership fuctio has a uique global maximum at x). We call u as a fuzzy real umber iduced by the real member x. A set of all fuzzy real umbers iduced by the real umber system the relatio ~ o F R as x ~ x if ad oly if x ad x are iduce same real umber x. The ~ is a equivalece relatio, which equivalece classes [x ] {a a~ x }. The quotiet set F R / ~ is the equivalece classes. The the cardiality of F R /~ is equal to the real umber system R sice the map R F R / ~ by x [x ] is Necall F R / ~ as the fuzzy real umber system. Fuzzy real umber system (F R / ~ ) R cosists of caoical fuzzy real umber we call F R / ~ ) R as the caoical fuzzy real umber system be a measurable space ad R, B be a Borel measurable space. (R) (Power set of R) be a set-valued fuctio. Accordig to is called a fuzzy-valued fuctio if {(x, y) : y f(x)} is M x B. f(x) is called a fuzzy-valued fuctio if f : X F (the set of all umbers). If f is a fuzzy-valued fuctio the f x is a set-valued fuctio [0, ]. f is called (fuzzy-valued) measurable if ad oly if f x is (set-urable for all [0,]. Make fuzzy radom variables more tractable mathematically, we strog sese of measurability for fuzzy-valued fuctios. f(x) be a closed-fuzzy-valued fuctio defied o X. From Wu wig two statemets are equivalet. f U (x) are (real-valued) measurable for all [0, ]. fuzzy-valued) measurable ad oe of f L x ad f U (x) is (real-value) measurable for all [0,]. A fuzzy radom variable called strogly measurable if oe of the above two coditios is easy to see that the strog measurability implies measurability. μ) be a measure space ad (R, B) be a Borel measurable space. (R) be a set-valued fuctio. For K R the iverse image of f. { x X f x K } u) be a complete σfiite measure space. From Hiai ad Umehaki ig two statemets are equivalet. Borel set K R, f (K) is measurable (i.e. f (K) M), y f x is M x B - measurable. If x is a caoical fuzzy real umber the x L x U, Let Xbe a fuzzy radom variable. x L ad x U are radom variables for all x ad x U. Let F(x) be a cotiuous distributio fuctio of a radom variable X. Let x L ad x U have the same distributio fuctio F(x) for all [0,]. For ay fuzzy observatio x of fuzzy radom variable X (X (ω) x ), the -level set x is x [x L, x U ]. We ca see that x L adx U are the observatios of x L ad x U, respectively. x L (ω) x L ad x U (ω) x U are cotiuous with respect to for fixed ω. Thus x L, x U is cotiuously shrikig with respect to. Sice [x L, x U ] is the disjoit uio of [x L, x L ] ad (x L, x U ] (ote that x L x U ), for ay real umber x [x L, x U ], we have x x β L or F (x β U ) with x. If we costruct a iterval 46 Page

3 Fuzzy radom variables ad Kolomogrov s A [mi { if β F(x β L ), if β F(x β L ) } max {sup β F(x β L ), sup β F(x β L )}] the this iterval will cotai all of the distributios. (values) associated with each of x [x L, x U ], We deote F x the fuzzy distributio fuctio of fuzzy radom variable X. The we defie the membership fuctio of F(x) for ay fixed xby ξ F(x) (r) sup 0 A, (r) via the form of Resolutio Idetity. we also say that the fuzzy distributio fuctio F(x) is iduced by the distributio fuctio F(x). Sice F(x) is cotiuous.we ca rewrite A α as A [mi { if β F(x β L ), mi β F(x β L ) } max {max β F(x β L ), max β F(x β L )}] I order to discuss the covergece i distributio for fuzzy radom variables i Sectio 4, we eed to claim F(x) is a closed-fuzzy-valued fuctio. First of all, we eed the followig propositio, We shall discuss the strog ad weak covergece i distributio for fuzzy radom variables i this sectio. We propose the followig defiitio. Defiitio 3. Let X ad {x } be fuzzy radom variables defied o the same probability space (Ω A P ). i) We say that {X } coverges i distributio to Xlevel-vise if (x ) α L ad (x ) α U coverge i distributio to X α L ad X α U respectively for all α. Let (x) ad F(x) be the respective fuzzy distributio fuctios of X α ad X. We say that {X α }coverges i distributio to X strogly if lim F x s F x ii) We say that {X } coverges i distributio to Xweakly if lim F x w F x From the uiqueess of covergece i distributio for usual radom variables.we coclude that the above three kids of covergece have the uique limits. 5. MAIN RESULTS THEOREM 5. (KOLMOGOROU CONVERGENCE THEOREM) Let {X} be idepedet fuzzy radom variables with EX 0 ad σ EX <, > If EX < the X coverges a.s. Proof : For > 0 P [ m k (0, [ ( (X k ) v (X k ] ] m k E ( (0,] [ ((X k ) v (X k ] < 3 if m, o Therefore 47 Page

4 Fuzzy radom variables ad Kolomogrov s P [ m k (0,] α [(X k v (X k ] ] i.e. P [ (0,] α [(S m (S V (S m ] ] if m, 0 i.e, {S } is a calculcy sequece i S P S Say. Hece by Levys Theorem S S a.s. i.e. X coveyes a.s. Defiitio 5.: Two sequeces of Fuzzy radom variables {X } ad {Y } are said to be tail equivalet if P (X Y ) < α THEOREM 5. : α Suppose that {X } be a sequece of idepedet fuzzy radom variables. Let X X if X 0 if X > for all The the series X coverges a.s. if the followig series coverges. (a) P (w : X w / > ) < (b) E (X ) ) coverges ad (c) σ X <α Proof Suppose that the three coditios hold. Because of bykolmogorov Khitchic theorem. [((X E(X ) v ((X E (X ) ] Coveys a.s. The (if) implies 48 Page

5 Fuzzy radom variables ad Kolomogrov s ((X v (X coverges a.s. By (a) ad Borel Ca telli lemma P ( (0,] [(X ) v (X ] > i.o. ) 0 So (0,] [(X ) v (X ] a.s. Thus (0,] [(X v (X ] Coverges a.s. Coversely if (0,] [(X v (X ] Coverges a.s. the the fuzzy radom variables X 0 a.s. Hece P ( (0,] ( X v (X ) > i.o. ) 0 This implies (a) byborel Zero oe law. Now {X } ad {X } are tail equivalet sequeces So it is clear that 0, [(X v (X ] coverges a.s. whe 0, [(X v (X ] coverges a.s. Now[ (0,] [(X ) v (X ] is a sequece of uiformly bouded idepdet fuzzy radom variables Let S j (0,] α ((X j v (X j ) Sice j (0,] α [(X (X )] coverges a.s. lim P( Sup m (0,] α [((S m (S ) V ((S m (S m ) ] 0 By the lower boud of Kolmogovovs iequality P (Sup m (0,] α [((S m (S ) V ((S m (S m ) Now if ( ) σ j X j 49 Page

6 Fuzzy radom variables ad Kolomogrov s σ Xj j, the we have P (Sup m (0,] α [((S m (S ) V ((S m (S m ) ) This cotradicts the cotetio () So j σ Xj < provig (c) The Khitchie Kolmogoroves theorem implies (0,] α ((X E(X )) V ((X E(X )) coverges a.s. Now Sice j X coverges a.s. We have (X ) coverget provig (b) THEOREM 5. 3 (KOLMOGOROVS INEQUALITY) Let X X X.... be idepedet fuzzy radom variables ad E(X i )<, i. If S i X i ad > 0 the a) P (max k (0,] α ((S k ) E(S k ) V ((S k E(S k ) k σ k ad if moreover (0,] α [ (X k ) (X k ) C < a.s. the b) ( ) P (max k (0,] α ( (S k ) E(S k σ k k V ((S k E(S k ) Proof : We assume EX k 0, k. Defie a fuzzy radom variables by st k, k otherwise such that S k if there is such a k. The (max k (0,] α [(S k V(S k ] ] [ t ] ad [t k] B(x x..x k ) Hece 50 Page

7 Fuzzy radom variables ad Kolomogrov s [tk] (0,] α [(S k ((S (S k ) V (S k ((S (S k )) d P E (0,] α [(S k I tk ((S (S k ) V ((S (S k )] E [ S k I [tk] E (((S (S k ) V ((S (S )] 0 Therefore, [tk] (S d P [tk] ((S k ((S (S k ) V ((S k ((S (S k ) d P [tk] ((S k ((S (S k ) V ((S k ((S (S ) ((S (S k ) (S k ) V (S (S k (S k ) d P [tk] (S k ) (S k ) ) d P P(t k) (5.) Therefore P (t ) P (t k) k [tk] (S k V (S ) d P [t ] (S k V (S ) d P (S V (S ) d P E (S ) ( 5.) But ES k E (X k V (X k ) Let X X X.... be idepedets. k σ k So from (3) ad () P [max k S k ] σ k k 5 Page

8 Fuzzy radom variables ad Kolomogrov s To prove the lower boud of Kolmogorovs iequality let f k I [t>k]. The f k S k ad X k are idepedet for k 0,, Now [t>k] [t k] c B, (x... x k ) sice [t k] B, (x... x k ). Therefore E (f k S k X k ) E (f k S k ) E (X k ) 0 Now E ((S k (f k E ((S k (f k V (S k (f k ) V (S k (f k ) E ((S k (f k V (S k (f k (X k (f k V (X k (f k ) E ((S k ) (f k V (S k (f k ) E ((X k (f k V (X k (f k ) E ((S k ) (f k V (S k (f k ) E ((X k V (X k E ((f k V (f k ) E ((S k ) (f k V (S k (f k ) E ((X k E (f k VE ((X k E(f k ) E ((S k ) (f k V (S k (f k ) E ((X k V (X k ) P (t > k-) (5.3) Agai E ((S k (f k V (S k (f k ) E ((S k (f k V (S k ) (f k ) E ((S k I [tk] V (S k ) I [tk] (5.4) 5 Page

9 Fuzzy radom variables ad Kolomogrov s From (5.3) ad (5.4) E ((S k ) (f k V (S k (f k ) E ((X k V (X k ) P (t > k-) E ((S k (f k V (S k ) (f k ) E ((S k (f k V (S k ) (f k E ((S k I [tk] V (S k ) I [tk] Sice (X k V (X k C for all K ad (X k E (X k V (X k E (X k C E ((S k ) (f k V (S k (f k ) E ((X k P (t k) V (X k ) P (t k) E ((S k (f k V (S k (f k ) ( C) P (t k) (5.5) Summig over (6) for k to ad after cacellatio we get k E ((X k ) V (X k ) ) P (t k) E ((S k (f V (S ) (f ) ( C) P (t ) Now S f By defiitio of t ad P(t > ) < P(t k) if k imply k E ((X k ) α P(t k) V E ((X k ) α P(t ) Or E ((f V (f ) ( C) P (t ) k E ((X k ) α V ((X k ) α P(t > ) P(t > ) ( C) P (t ) ( C) P (t ) ( C) P (t > ) 53 Page

10 Fuzzy radom variables ad Kolomogrov s ( C) Hece P (t ) ( C) / k E ((X k ) α V ((X k ) α ( C) σ k k implies P (t ) ( C) / σ k LEMMA 5. : (KRONECKER S LEMMA) For sequeces {a } ad {b } of fuzzy real umbers ad a coverges ad b b k b k a k 0 as Proof : Sice a coverges S a k S (say) b k [(b k (a k V (b k (a k b k [(b k ((S k (S k (b k ((S k (S k )] b ( (b k (S k V(b k (S k (b k (S k V(b k (S k ) b ( (b k (S k V(b k (S k (b k (S k V(b k (S k ) b (((b (S V(b (S ( (b k (b k (S k ) V ((b k (b k ) (S k (S V(S b ((b k (b k ) (S k ) V ((b k (b k ) (S k 54 Page

11 Fuzzy radom variables ad Kolomogrov s S S 0 Sice ( b (b k (b k ((S k (S ) V ((b k (b k ) ((S k (S ) (S ( (b ) (b k (b k V (S (b ) (b k (b k Now [ b (b k (b k ](S k V [ (b (b k (b k ](S k (S (b ) (b k (b k V (S (b ) (b k (b k (S (b ) ((b (b ) V (S (b ) ((b (b ((S V (S ) as b ad ( (b ) (b k (b k )( S k (S k V (b ( ) 0 as (b k (b k )((S k (S Sice ( (b ) (b k (b k )( S k (S k 55 Page

12 V (b ( ) (b k (b k )((S k (S ) Fuzzy radom variables ad Kolomogrov s 0 k ( (b k (b k ) S k (b ) (S k 0 V (b ( ) k b k (b k )((S k (S ) 0 ( (b k (b k ) ( S k (b ) (S k V (b ( ) 0 b k (b k )((S k (S ) for > 0 (b 0 (b ) (b ) V (b 0 (b ) (b ) if > 0 LEMMA 5. : (Loeve) Let X be a fuzzy radom variables ad q(t) P { X V X > t } F(t) F (t) For every y > 0, x > 0 we have x r q ( l/r x ) E ( X r V X r X r V X r q ( l/r x ) Proof : E ( X r V X r 0 d P ( X V X r t ) 0 t r dq (t) /r x ( ) /r x) t r dq (t), x > 0 Now rx ( ) rx t r dq (t) x r [ q rx q ( rx)] 56 Page

13 Fuzzy radom variables ad Kolomogrov s ad rx ( ) rx t r dq (t) (-)x r [ q rx q ( rx)] If E ( X r v X r ) the proof is obvious. ad if E ( X r v X r ) < the x r Nq (N rx) 0 as N I fact, > E ( X r v X r ) E ( X r v X r ) I ( X v X > x N r] Nx r P [ X r v X r >N x ] Nx r q [ N r x ] If E ( X r v X r ) < by absolute cotiuity. of itegral Nx r q [ N r x ] 0 as N o the other had. E ( X r v X r ) < ( ) x r [ q ( ) rx q ( r )x ] Sice N x r [ q () rx ] (N)x r q ( r )x Nq ( rx) 0 the right had side of the last iequality teds to x r q ( r x ) Now if q ( r x )< the q ( r x ) 0 ad 57 Page

14 Fuzzy radom variables ad Kolomogrov s E ( X r v X r ) lim N N x v [ q ( ) rx q ( r x) ] lim N x v ( N q ( rx ) Nq ( r x) ) x v ( q ( rx )) which completes the proof. THEOREM 5.4: (KOLMOGOROVS STRONG LAW OF LARGE NUMBERS for idepedet ideticals distributed r.v.s.) Let {x } be a sequece of idedet idetically distributed fuzzy radom variables the S C < a.s. if ad oly if E ( (X ad the C E (X ) v (X ) < Proof For the oly if part let A ( X v X ) the E ( (X v (X P A (5.7) Now P(A) P ( X v X ) P ( (X v (X ) S a.s. < the (X V (X (S V (S ( ) (S ) (S ) C C 0 a. s. Hece P ( (X V (X > i.o. ) By Borel 0 Law P ( (X v (X ) i.e. P ( (X v (X ) ) < 58 Page

15 Fuzzy radom variables ad Kolomogrov s > P [ (X v (X ) ) P (A ) P (A ) < So from () E ( (X v (X ) ) < Coversely let E ((X v (X ) < ad C E ((X v (X ) Defie (X k V (X k ((X v (X ) I [ X k k ] k,,3,... ad (S V (S ( X v X X v X.... X v X The X k, k,,.... are idepedet ad (X k ) v (X k k Now k P [ (X k v (X k (X k v (X k P ( (X k v (X k > k) P (A k ) P [ (X k v (X k (X k v (X k i.o. Hece S ad S treds to the same limit a.s. if they coverge at all i. (S v (S (S v (S 0 a. s. as 59 Page

16 Fuzzy radom variables ad Kolomogrov s So it is eough to prove that S E ((X v (X ) < Now X are idepedet but may be ecessarily be idetically distributed, we shall show that σ ((X ) v (X ) < ad that will imply S E (S ) coverges to zero almost surely. E ( X v X ) E ( X v X ) I [ X v X ) ] E ( X v X ) I [ X v X ) ] Therefore E ( X v X ) G.S. E ( (S V (S ) E ( X v X ) σ ( (X V (X ) ) E ( (X ) V (X ) ) [ X ((X ) V (X ) ) dp v X ] k [k < X ((X ) V (X ) ) dp v X ] k k [k < X ((X ) V (X ) ) dp v X ] k k k [k < X ((X ) V (X ) ) dp v X ] k k k k P [k < X v X ] k ] 60 Page

17 Fuzzy radom variables ad Kolomogrov s k k P [ (k ) < X v X < k ] k (k)p [ k < ( X v X k ] [k < ( X ((X ) V (X ) ) v X ] k ( E ( X v X REFERENCES [] A.Colubi, M. Lopez Diaz, J.S. Domigueez Mecheru, M.A. Gil, A Geeralized strog law of large umbers Pro theory. Relat. Field 4- (999) [] E.P. Klemet, M.L. Puri, P.A.Ralesuv, Limit theorems for Fuzzy Radom Variables, Proce. Roy. Soc. Lodo Set. A. 407(986) 7-8. [3] H. Jouse, A Strog law of large umbers for fuzzy radom sets, Fuzzy sets ad Systems 4(99) [4] H. Kwakerack, Fuzzy radom variables I Defiitios ad theorems, Iform Sai. 5 (978) -9. [5] M. Lopez-Diaz, M.Gil, Approximatig itegrally laded fuzzy radom variables i terms of the gerealised Hosdorff Matric. Iform Sei. 04 (998) [6] M.L. Puri, D.A. Ralese, Fuzzy Radom variables J. Maths Aas. Appl. 4(986) [7] M.L. Puri D.A. Rabise, Fuzzy Radom Variables, J.Math Aal. Appl. 4(986) [8] R. Kroer, o the variace of fuzzy radom variables Fuzzy Seb ad systems 9 (997) [9] Y.Feg, Covergece theorems for fuzzy radom variables ad fuzzy martigales, Fuzzy sets ad systems 03 (999) [0] Y. Feg, Mea-Square itegral ad differetial of Fuzzy stochastic processes, Fuzzy sets ad systems 0 (999) [] Y. Feg, Decompositio theorems for fuzzy super martigales ad sub martigales, Fuzzy sets ad systems, 6 (000) [] Y. Feg, L.Hu, H. Shu, The variace ad covariace of fuzzy radom variables ad their applicatios, fuzzy sets ad systems 0(00) Page