Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in Gα Space

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1 Advaces Pure Matheatcs Publshed Ole August 26 ScRes Strog Laws of Large Nubers for uzzy Set-Valued Rado Varables G Space Lae She L Gua College of Appled Sceces Beg Uversty of Techology Beg Cha Receved 7 July 26; accepted 2 August 26; publshed 5 August 26 Copyrght 26 by authors ad Scetfc Research Publshg Ic Ths wor s lcesed uder the Creatve Coos Attrbuto Iteratoal Lcese (CC BY) Abstract I ths paper we shall preset the strog laws of large ubers for fuzzy set-valued rado varables the sese of d The results are based o the result of sgle-valued rado varables obtaed by Taylor [] ad set-valued rado varables obtaed by L Gua [2] Keywords Laws of Large Nubers uzzy Set-Valued Rado Varable ausdorff Metrc Itroducto Wth the developet of set-valued stochastc theory t has becoe a ew brach of probablty theory Ad lts theory s oe of the ost portat theores probablty ad statstcs May scholars have doe a lot of research ths aspect or exaple Artste ad Vtale [3] had proved the strog law of large ubers for depedet ad detcally dstrbuted rado varables by ebeddg theory a [4] had exteded t to separable Baach space Taylor ad Ioue had proved the strog law of large ubers for depedet rado varable the Baach space [5] May other scholars also had doe lots of wors the laws of large ubers for set-valued rado varables I [2] L proved the strog laws of large ubers for set-valued rado varables G space the sese of d etrc As we ow the fuzzy set s a exteso of the set Ad the cocept of fuzzy set-valued rado varables s a atural geeralzato of that of set-valued rado varables so t s ecessary to dscuss covergece theores of fuzzy set-valued rado sequece The lts of theores for fuzzy set-valued rado sequeces are also bee dscussed by ay researchers Colub et al [6] eg [7] ad Molchaov [8] proved the strog laws of large ubers for fuzzy set-valued rado varables; Pur ad Ralescu [9] L ad Ogura [] proved cover- ow to cte ths paper: She LM ad Gua L (26) Strog Laws of Large Nubers for uzzy Set-Valued Rado Varables G Space Advaces Pure Matheatcs

2 L M She L Gua gece theores for fuzzy set-valued artgales L ad Ogura [] proved the SLLN of [2] the sese of d by usg the sadwch ethod Gua ad L [3] proved the SLLN for weghted sus of fuzzy setvalued rado varables the sese of d whch used the sae ethod I ths paper what we cocered are the covergece theores of fuzzy set-valued sequece G space the sese of d The purpose of ths paper s to prove the strog laws of large ubers for fuzzy set-valued rado varables G space whch s both the exteso of the result [] for sgle-valued rado sequece ad also the exteso [2] for set-valued rado sequece Ths paper s orgazed as follows I Secto 2 we shall brefly troduce soe cocepts ad basc results of set-valued ad fuzzy set-valued rado varables I Secto 3 I shall prove the strog laws of large ubers for fuzzy set-valued rado varables G space whch s the sese of ausdorff etrc d 2 Prelares o Set-Valued Rado Varables Throughout ths paper we assue that ( Ω µ ) s a coplete probablty space ( ) Baach space K ( ) s the faly of all oepty closed subsets of ad b( ) ( ( ) ) of all o-epty bouded closed(copact) subsets of ad ( ) s a real separable K K s the faly K c s the faly of all o-epty copact covex subsets of Let A ad B be two oepty subsets of ad let λ the set of all real ubers We defe addto ad scalar ultplcato by The ausdorff etrc o K ( ) s defed by { : } A+ B= a+ b a Ab B { λ : } λa= a a A ( ) = ax { supf supf b B a A } d A B a b a b a A for AB K ( ) or a A K ( ) let A = d ({ } A K ) The etrc space ( K b( ) d ) s coplete ad K bc ( ) s a closed subset of ( b( ) d ) b B K (cf [4] Theores 2 ad 3) or ore geeral hyperspaces ore topologcal propertes of hyperspaces readers ay refer to the boos [5] ad [4] A K defe the support fucto by or each s x * A = sup x * a x * * * where s the dual space of * * * Let S deote the ut sphere of C ( S ) the all cotuous fuctos of as v = sup C x S The followg s the equvalet defto of ausdorff etrc AB K bc or each a A * * * * { } d A B = sup s x A s x A : x S A set-valued appg : fucto) f for each ope subset O of ( O) { ω : ( ω) O } or each set-valued rado varable the expectato of deoted by [ ] * S ad the or s defed Ω K s called a set-valued rado varable (or a rado set or a ult- = Ω E s defed by [ ] = { µ } E fd : f S Ω where f d µ s the usual Bocher tegral L [ Ω ] the faly of tegrable -valued rado varables Ω S = f L Ω; : f ω ω ae µ { } ad [ ] 584

3 L M She L Gua Let ( ) deote the faly of all fuctos v : [ ] whch satsfy the followg codtos: ) The level set v = { x : v( x) = } 2) Each v s upper secotuous e for each ( ] the level set v = { x : v( x) } closed subset of 3) The support set v = cl { x + : v ( x) > } s copact A fucto v ( ) s called covex f t satsfes v( λx+ ( λ) y) { v( x) v( y) } s a for ay xy λ ( ] Let c ( ) be the subset of all covex fuzzy sets ( ) It s ow that v s covex the above sese f ad oly f for ay ( ] subset of (cf Theore 32 of [6]) or ay v ( ) the closed covex hull cov c cov = for all ( ] defed by the relato or ay two fuzzy sets 2 cov ν ν defe ( ] { } ν + ν 2 x = sup : x ν 2 + ν for ay x Slarly for a fuzzy set ν ad a real uber λ defe for ay x The followg two etrcs ( ) ad [8] or [4]): for v v 2 ( ) Deote x The space ( ) ( λν )( x) = { ( ] x λν } sup : the level set v s a covex of v s whch are extesos of the ausdorff etrc d are ofte used (cf [7] ( ) d v v = sup d v v 2 2 ( ] d v v = d v v d 2 2 v = : d vi = sup > v where I K s the fuzzy set tag value oe at ad zero for all ( d ) s a coplete etrc space (cf [8] or [4]: Theore 56) but ot separable (cf [7] or [4]: Rear 57) It s well ow that v v d every Cauchy sequece { } ( ) = for every ( ] < Due to the copleteess of v : has a lt v ( ) A fuzzy set-valued rado varable (or a fuzzy rado set or a fuzzy rado varable lterature) s a appg : Ω ( ) such that ( ω) = { x : ( ω)( x) } s a set-valued rado varable for every ( ] (cf [8] or [4]) The expectato of ay fuzzy set-valued rado varable deoted by E[ ] s a eleet ( ) such that for every ( ] ( E[ ] ) E[ ] = where the expectato of rght had s Aua tegral ro the exstece theore (cf [9]) we ca get a equvalet defto: for ay x Note that E[ ] s always covex whe ( µ ) 3 Ma Results = { [ ] [ ]} E x sup : x E Ω s oatoc I ths secto we wll gve the lt theores for fuzzy set-valued rado varables G space I wll frstly 585

4 L M She L Gua troduce the defto of G space The followg Defto 3 ad Lea 32 are fro Taylor s boo [8] whch wll be used later Defto 3 A Baach space s sad to satsfy the codto G for soe ( ] If there exsts a appg G : such that ) G( x) 2) = x ; G x x x + = ; 3) G( x) G( y) = Ax y for all xy ad soe postve costat A Note that lbert spaces are G wth costat A = ad detty appg G Lea 32 Let be a Baach space whch satsfes the codto of G { V V2 V} be depedet E V < + for each = 2 The rado eleets such that EV [ ] = ad = E V V A E V where A s the postve costat 3) of defto 3 I order to obta the a results we frstly eed to prove Lea 35 The followg lea are fro [4] (cf p89 Lea 34) whch wll be used to prove Lea 35 C : N K If Lea 33 Let { } be a sequece for soe C K ( ) c the Lea 34 (cf [3]) or ay ν ( ) l d coc C = = l d C C = = there exsts a fte ( ) d ν ν + ε for all = M t t Now we prove that the result of Lea 33 s also true for fuzzy sets : If Lea 35 Let { } for soe ν ( ) c ν be a sequece the Proof By (3) we ca have = t < t < < t M = such that l d coν ν = (3) = l d ν ν = = ad l d coν ν = = for ( ] The by Lea 33 for ( ] l d coν+ ν+ = = we have 586

5 L M She L Gua ad l d ν ν = = l d ν+ ν+ = = By Lea 34 tae a ε > there exsts a fte = < < < M = such that The for < < Cosequetly d ( ) ν ν ε for all M + d ν ν d ν ν d ν ν + = + = = d + d + d ν ν 2 ν ν ν ν = = sup ν ν ax ν ν + ax ν ν + 2 ε d d ( ] d M M + + = = = Sce the frst two ters o the rght had coverge to probablty oe we have but ε s arbtrary ad the result follows l sup sup d ν ν 2 ε ( ] = Theore 36 Let be a Baach space whch satsfes the codto of G let { } : be depe- for ay If det fuzzy set-valued rado varables where = for t ad t t of d Proof Defe Note that such that E = I = ( ) E < + t = t for t the U = I W = I { } { > } = + for each ad both { W : } ad { : } W U fuzzy set-valued rado varables Whe > we have for ay E W E W E coverges wth probablty the sese = W U are depedet sequece of = ad = = = = = The Ad fro E < we ow that = E W : = s a Cauchy sequece So we have 587

6 L M She L Gua E W coverges as = Sce covergece the ea pled covergece probablty Ito ad Nsos result [9] for depedet rado eleets (cf Secto 45) provdes that = So for ay > by tragle equalty we have It eas we have W coverges probablty d W W = d W W + W = = = = = + d W W + d I W = = = + = d I W = + = W ae as = + W : s a Cauchy sequece the sese of = W coverges alost everywhere the sese of = Next we shall prove that d U coverges the sese of = ( d ) d By the copleteess of ( ) d rstly we assue that { } fuzzy set-valued rado varables The by the equvalet defto of ausdorff etrc we have or ay fxed there exsts a sequece E U = E sup U = ( ] = K = E sup d U { } ( ] = = E sup sup s x U ( ] x S = x S such that l s x U = sup s x U = x S = U are all covex That eas there exst a sequece x S such that E U = E sup l s x U ( ] = = The by Cr equalty doated covergece theore ad Lea 32 we have 588

7 L M She L Gua E U = E sup l s x U ( ] = = for each ad E sup l s( x U ) ( ] = E l sup s( x U ) ( ] = l E sup s( x U ) Es( x U ) + Es( x U ) ( ] = = l E l s( x U ) Es( x ) ( ) U + Es x U = = l l E s( x U ) Es( x ) ( ) U + Es x U = l l E s x U E s x U + E s x U = = ( ) ( ) ( ) l l 2 E s( x U ) Es( x ) ( ) U + Es x U = = l l 2 AE s( x U ) E s( x ) ( ) U Es x U + = = l l 2 2 A E s( x U ) 2 A E s( x ) ( ) U Es x W + + = = = l l 2 2 A E s x U 2 A E s x U E + + = = = ( ) ( ) s( x W ) 2+ = l l 2 2 A E s( x U ) E s( x ) W + = = 2+ l 2 2 AE sup s( x U ) E sup s( x W ) + * * * * = x S = x S = l 2 2 A E U + E W K K = = = 2 2 A E U + E W = = = = A E ( ) E ( ) 589

8 L M She L Gua The we ow E = U Thus by the slar way as above to prove ca prove that wth probablty the sese of So we ca prove that wth probablty the sese of cou coverges wth proba- = blty the sese of d s a Cauchy sequece ece { E U = } s a Cauchy sequece W coverges wth probablty the sese of = = U coverges d I fact for each d U U = d U U + U = = = = = + d If { } = U = + ae as coverges U are ot covex we ca prove as above ad by the Lea 35 we ca prove that d We also U coverges wth proba= blty the sese of d The the result was proved ro Theore 36 we ca easly obta the followg corollary Corollary 37 Let be a separable Baach space whch s G for soe < Let { : } be a sequece of depedet fuzzy set-valued rado varables ( ) such that E = I for each If + + ( t) t : = 2 are cotuous ad such that ad are o-decreasg the for each t t + the covergece of ples that E ( ) ( ) = coverges wth probablty oe the sese of d Proof Let U = I ad W = I > If > by the o-decreasg property of = { } { } ( t) t we have ( ) ( ) 59

9 L M She L Gua That s ( ) (4) If by the o-decreasg property of t ( t) we have That s ( ) ( ) + (42) The as the slar proof of Theore 36 we ca prove both oe the sese of d ad the result was obtaed U ad = W coverges wth probablty = Acowledgeets We tha the Edtor ad the referee for ther coets Research of L Gua s fuded by the NSC ( ) Refereces [] Taylor RL (978) Lecture Notes Matheatcs Sprger-Verlag 672 [2] L G (25) A Strog Law of Large Nubers for Set-Valued Rado Varables G Space Joural of Appled Matheatcs ad Physcs [3] Artste Z ad Vtale RA (975) A Strog Law of Large Nubers for Rado Copact Sets Aals of Probablty [4] a (984) Strog Laws of Large Nubers for Multvalued Rado Varables Multfuctos ad Itegrads I: Salett G Ed Lecture Notes Matheatcs Vol 9 Sprger Berl 6-72 [5] Taylor RL ad Ioue (985) A Strog Law of Large Nubers for Rado Sets Baach Spaces Bullet of the Isttute of Matheatcs Acadea Sca [6] Colub A López-Díaz M Doguez-Mechero JS ad Gl MA (999) A Geeralzed Strog Law of Large Nubers Probablty Theory ad Related elds [7] eg Y (24) Strog Law of Large Nubers for Statoary Sequeces of Rado Upper Secotuous uctos Stochastc Aalyss ad Applcatos [8] Molchaov I (999) O Strog Laws of Large Nubers for Rado Upper Secotuous uctos Joural of Matheatcal Aalyss ad Applcatos [9] Pur ML ad Ralescu DA (99) Covergece Theore for uzzy Martgales Joural of Matheatcal Aalyss ad Applcatos [] L S ad Ogura Y (23) A Covergece Theore of uzzy Valued Martgale the Exteded ausdorff Metrc uzzy Sets ad Systes [] L S ad Ogura Y (23) Strog Laws of Nubers for Idepedet uzzy Set-Valued Rado Varables uzzy Sets ad Systes [2] Ioue (99) A Strog Law of Large Nubers for uzzy Rado Sets uzzy Sets ad Systes [3] Gua L ad L S (24) Laws of Large Nubers for Weghted Sus of uzzy Set-Valued Rado Varables Iteratoal Joural of Ucertaty uzzess ad Kowledge-Based Systes

10 L M She L Gua [4] L S Ogura Y ad Kreovch V (22) Lt Theores ad Applcatos of Set-Valued ad uzzy Set-Valued Rado Varables Kluwer Acadec Publshers Dordrecht [5] Beer G (993) Topologes o Closed ad Closed Covex Sets Matheatcs ad Its Applcatos Kluwer Acadec Publshers Dordrecht ollad [6] Che Y (984) uzzy Systes ad Matheatcs uazhog Isttute Press of Scece ad Techology Wuha (I Chese) [7] Kleet EP Pur LM ad Ralescu DA (986) Lt Theores for uzzy Rado Varables Proceedgs of the Royal Socety of Lodo A [8] Pur ML ad Ralescu DA (986) uzzy Rado Varables Joural of Matheatcal Aalyss ad Applcatos [9] L S ad Ogura Y (996) uzzy Rado Varables Codtoal Expectatos ad uzzy Martgales Joural of uzzy Matheatcs Subt or recoed ext auscrpt to SCIRP ad we wll provde best servce for you: Acceptg pre-subsso qures through Eal aceboo LedI Twtter etc A wde selecto of ourals (clusve of 9 subects ore tha 2 ourals) Provdg 24-hour hgh-qualty servce User-fredly ole subsso syste ar ad swft peer-revew syste Effcet typesettg ad proofreadg procedure Dsplay of the result of dowloads ad vsts as well as the uber of cted artcles Maxu dsseato of your research wor Subt your auscrpt at: 592