Reseach ad Coucatos atheatcs ad atheatcal ceces Vol 9 Issue 7 Pages 37-5 IN 39-6939 Publshed Ole o Novebe 9 7 7 Jyot cadec Pess htt//yotacadecessog UBEQUENCE CHRCTERIZT ION OF UNIFOR TTITIC CONVERGENCE OF DOUBE EQUENCE FIKRET ČUNJO ad FTIH DETOVIĆ Deatet of atheatcs Faculty of Natual ceces ad atheatcs Uvesty of aaevo Bosa & Hezegova e-al fcualo@hs-hbba fath_d@sco bstact I ths ae t s show that alost evey tes of P subseuece of double seuece s ot ufoly statstcally coveget to f coveges to ufoly statstcally lost evey tes of easue of double seuece coveges to the Pgshe s sese f coveges to ufoly statstcally ad dvegetly the Pgshe s sese Ths s ot tue fo P P subseuece Itoducto The cocet of the statstcal covegece of a seueces of eals was toduced by Fast [3] Futheoe Göha et al [6] toduced the oto of otwse ad ufo statstcal coveget of double seueces of eal-valued fucto Çaa ad ltay [5] eseted ult- atheatcs ubect Classfcato Pay 4B5; ecoday 435 4G5 Keywods ad hases double seueces ufo statstcal covegece Receved Octobe 6 7; Revsed Novebe 8 7
38 FIKRET ČUNJO ad FTIH DETOVIĆ desoal aalogues of the esults eseted by Fdy lle ad Oha [4 5 7] Düda ad tay [6-] vestgated the elato betwee I-covegece of double seueces Now we ecall that the deftos of cocets of deal covegece ad basc cocets [ 8] The seuece of eal ubes coveges to the Pgshe s sese f fo K such that K We wte l et K N N et K be the ube of ( ) K such that If d ( K ) l K the Pgshe s sese The we say that K has double atual desty et s seuece of eal ubes ad et () {( ) N N } The seuece fo statstcally coveges to R f d ( () ) We wte st l et s set X / class I of subsets of X s sad to be a deal X ovded the followg stateets hold () / I; () B I B I; () I B B I
UBEQUENCE CHRCTERIZTION OF 39 I s otval deal f X I otval deal I s called adssble f { } I fo X I ths ae the focus s ut o deal belogs to the I u f N N u I defed by subset l { ( ) } ufoly o set N the Pgshe s sese That s subset of the N N s ufoly statstcally desty zeo The seuece ufoly statstcally coveges to f fo ay That s seuece K such that {( ) N N } I u ufoly statstcally coveges to f { } K N We wte Ust l We deote wth X a set of all double seueces of s ad s e X { { } N} et seuece ad X The wth we deote a seuece defed followg way fo whch we efe to as subseuece of seuece The ag s a becto of the set X to a set of all subseueces of the seuece
4 FIKRET ČUNJO ad FTIH DETOVIĆ The ude the ebesgue easue o the set of all subseueces of the seuece cosde ebesgue easue o the set X et β sallest subsets the fo of σ -algeba subsets of the set X whch cotas of { ( ) X a a a } a a { } N I [3] t was ove that thee s a uue ebesgue easue P o the set X fo whch the followg ales P({ ( ) X }) a a a New Results lost evey double seueces of s ad s s ot alost coveget [4] Ths aalogue s vald also fo ufo statstcal covegece Theoe lost evey double seuece of s ad s s ot ufoly statstcally coveget Poof et { X } ce P ( ) N t s P( ) ce ae deedet based o the secod at of Boel-Catell lea P l su
UBEQUENCE CHRCTERIZTION OF 4 We deote l su The P Fo N thee est a bloc coosed of oes It follows does ot covege ufoly statstcally to We deote B { X } B l su B B B Coletely aalogously we coclude that P ( B) Fo B N thee ests a bloc coosed of zeos It follows B does ot covege ufoly statstcally to Evey ufoly statstcally coveget seuece to o The X coveges { X ( ) coveget ufoly statstcally} { X Ust l } { X Ust l } c c c B ( B ) It follows P({ X ( ) coveget ufoly statstcally}) P( B ) Defto The subseuece of seuece ufoly statstcally coveges to f K such that fo K ad N ovded that we have { } { } We wte Ust l
4 FIKRET ČUNJO ad FTIH DETOVIĆ lost evey subseuece of statstcally coveget double seuece s statstcally coveget The aalogue does ot aly to ufoly statstcally covegece Theoe 3 et Ust l ad the seuece s dveget the Pgshe s sese The Poof et ({ X Ust l }) P Tuv { X u( u ) v( v ) Due to the dvegece of the seuece N N N such that T uv / fo ftely ( u v) The P ( T uv ) fo ftely ( u v) T uv ae deedet ad P ( Tuv ) ( u v) T / ( ) / uv u v T uv Due to secod at of Boel-Catell lea t follows P( T ) Hece P T P T P l su T uv ( u v) T / uv et T The N N N such that { X } { }
UBEQUENCE CHRCTERIZTION OF 43 Hece does ot covege to ufoly statstcally o ({ X Ust l }) P Eale et chaactestc N N ufoly desty zeo wth the followg N ( ) such that et the seuece ( ) defed as ( ) ( ) The N the followg s vald { } { ( ) } fo Resectvely Ust l et B { X ( ) } et s the fte seuece ( ) such that ad ( ) The P ({ X }) Due to secod at of Boel-Catell lea P ( B) ce subseuece of does ot covege to the Pgshe s sese f ad oly f B t P({ X l the Pgshe s sese})
44 FIKRET ČUNJO ad FTIH DETOVIĆ et N N Thee s a uue easue P o X wth the oety P ({ X }) ( ) ( ) P ({ X a a }) P ({ X a }) P ({ X a }) alogue theoe s vald et the seuece ( ) be dveget the Pgshe s sese The statstcally coveges to N N wth desty zeo such that P ({ X l the Pgshe s sese}) Theoe 4 et the seuece ( ) dveget the Pgshe s sese The ufoly statstcally coveges to N N ufoly desty zeo such that P ({ X l the P gshe s sese}) Poof Because of ea the followg s vald Ust l N N ufoly desty zeo such that the subseuece ( y) of coveges to the Pgshe s sese fo y ( ) ( ) Not geealzg we ca assue that s ot a ot accuulato of the subseuece fo ( ) ( )
UBEQUENCE CHRCTERIZTION OF 45 Hece the subseuece z coveges to the Pgshe s sese f ad oly f N such that { } / z N N et { } B B X B The N s B P B P Hece B P ce the set B s a set of all X fo whch does ot covege to the Pgshe s sese It ({ }) sese s Pgshe' the l X P et the seuece ot be ufoly statstcally coveget ad let N N abtay ufoly desty zeo The due to the lea the subseuece s dveget the Pgshe s sese fo The followg cases ca be eseted (a) fo (b) fo (c) λ µ
46 FIKRET ČUNJO ad FTIH DETOVIĆ It follows (a) P ({ X }) (b) P ({ X }) (c) P ({ X }) P ({ X }) The due to secod at of Boel-Catell lea the followg s vald (a) P ({ X fo fte }) (b) P ({ X fo fte }) (c) P ({ X fo fte }) It follows Hece P P ({ X dveget the Pgshe s sese}) ({ X coveget the Pgshe s sese}) ea 5 Ust l N N ufoly desty zeo such that l the Pgshe s sese fo ( ) ( ) Poof et Ust l The thee s a seuece of atual ubes ( ) such that et N
UBEQUENCE CHRCTERIZTION OF 47 We defe X the followg way Fo N such that Fo the defto of the seuece we have N Hece the subseuece coveges to the Pgshe s sese Fo N such that et the { } { } { } { }
FIKRET ČUNJO ad FTIH DETOVIĆ 48 N { } N Hece such that { } N Resectvely s ufoly desty zeo We assue that thee s a subset of set N N ufoly desty zeo such that subseuece of coveges to the Pgshe s sese fo The N such that { } { } { }
UBEQUENCE CHRCTERIZTION OF 49 { } { ( ) } { } { ( ) } Obvously the fst suad coveges to zeo ufoly o N The secod suad coveges to zeo ufoly o N due to the assuto o Ust l Refeeces [] B ltay ad F Başa oe ew saces of double seueces J ath al l 39() (5) 7-9 [] Balceza K Des ad Kosas tatstcal covegece ad deal covegece fo seueces of fuctos J ath al l 38 (7) 75-79 [3] Cac F Čualo ad H I lle ubseuece chaactezatos of statstcal covegece of double seueces Rad at (4) 63-75 [4] F Čualo lost covegeces of double subseueces Float (8) 87-93 [5] C Çaa ad B ltay tatstcally boudedess ad statstcal coe of double seueces J ath al l 37 (6) 69-697 [6] E Düda O ough I -covegece of double seueces uecal Fuctoal alyss ad Otzato 37(4) (6) 48-49 [7] E Düda ad B tay I -covegece of double seueces of fucto Electo J ath al l 3() (5) - [8] E Düda ad B tay I -covegece ad I -Cauchy of double seueces cta ath c e B Egl Ed 34 B() (4) 343-353 [9] E Düda ad B tay I -ufo covegece of double seueces of fucto Float 3(5) (6) 73-8 [] E Düda ad B tay O soe oetes of I -covegece ad I -Cauchy of double seueces Ge ath Notes 7() () -
5 FIKRET ČUNJO ad FTIH DETOVIĆ [] E Düda ad C Çaa Rough covegece of double seueces Deostato atheatca 47 (4) 638-65 [] E Düda ad Y eve ultles fo bouded statstcal covegece of double seueces It ath Fou 7(5) () 58-587 [3] H Fast u la covegec statstue Collog ath (95) 4-44 [4] J Fdy O statstcal covegece alyss 5 (985) 3-33 [5] J Fdy ad C Oha tatstcal lt sueo ad feo Poc e ath oc 5 (997) 365-363 [6] Göha Gügö ad Et tatstcal covegece of double seueces of eal-valued fucto It ath Fou (8) (7) 365-374 [7] H I lle ad C Oha O alost coveget ad statstcally coveget subseueces cta ath Hug 93 () 35-5 [8] E Tas ad T Yudaad Chaactezato of ufo statstcal covegece fo double seueces solc atheatcal Notes 3 () 543-553 g