Almost Sure Convergence of Pair-wise NQD Random Sequence

Transcription

1 Moder Appled Scece Vol. 4 o. ; December 00 Almost Sure Covergece of Par-wse QD Radom Sequece Yachu Wu College of Scece Gul Uversty of Techology Gul Cha Tel: E-mal: wyc@glte.edu.c Dawag Wag College of Scece Gul Uversty of Techology Gul Cha Tel: E-mal: ytcdw@63.com Ths wor was partally supported by the Guagx atoal Scece Foudato of Cha o. 00GXSFA03) ad the Gul techology Foudato of Guagx Gul o. [009]3) Abstract I ths paper cosderg the sequeces of parwse QD that s appled broadly through troducg the slowly varyg fucto we exted a seres of coclusos. Keywords: Parwse QD radom sequeces Almost sure Slowly varyg fucto. Itroducto ad Lemmas Defto Lehma E L. 966): Let radom varables X ad Y are sad to be QD egatvely Quadrat Depedet) f x y R PX xy y) PX xpy ) y). Where X ad Y are sad to be QD egatvely Quadrat Depedet) A sequece of radom varablesx ; s sad to be parwse QD. Ths defto was troduced by Lehma. Obvously Parwse QD radom sequeces cotas a d of egatvely correlatve sequeces of A LQD D radom varables. For A radom sequeces a umber of wrters have obtaed as same as the covergece property may depedet codtos the propertes of lmt behavor of LQD D sequeces seldom appear lterature. Matula Matula P. 99) obtaed strog law of large umbers for Kolmogorov as same as them uder the depedet the dstgushg theorem of the Complete Covergece of Baum ad KataWAG Y B Chu Su & Xuguo Lu. 998) was obtaed by author Yuebao Wag who cosdered * Quyg Wu WU Q Y. 005) obtaed the wea law of large umbers ad the crtera theorem of Complete Covergece of Baum ad Kata WU Q Y. 00) o the codato that parwse QD. These propertes acheved the results of depedece codto ad the results of propertes of JamsoJamso B O Rey S & Prutt W. 965) Weghted Sums s obtaed. Curretly a umber of wrters have studed a seres of useful results of the lmt of parwse QD radom Sequece. Yuebao Wag WAG Y B YA J-gao & CHEG Feg-yag etal. 00) studed the strog stablty of dfferet dstrbuto parwse QD Wacheg GaoWA Cheg-gao. 005) studed f the wea law of large umbers of parwse QD ad codto of ra Cesàro s uformly tegrablty s the covergece propertes of Lr ad Yapg CheCHE Pg-ya. 008) studed the covergece propertes of Lr satsfed parwse QD the codto of uformly tegrablty s r r ) Cesàro s uformly tegrablty as same as them uder the codto of depedet ad the teacher Yuebao Wag WagWAG Y B YA J G & CAI X Z. 00) studed dfferet dstrbutos strog stablty of par-wse QD ad Quyg WuWU Q Y. 00) studed the three seres theorem of par-wse QD. I the followg Almost Sure Covergece of QD par-wse radom Sequeces are exteed o the codato that slowly varyg fucto. Defto WU Q Y. 00): Let l x be a Postve Fucto x [0 ) whch satsfes x the Publshed by Caada Ceter of Scece ad Educato 93

2 Moder Appled Scece Vol. 4 o. ; December 00 lcx lm for all c 0. x lx For postve fuctos as we ow f there exst a Postve Fucto such as l x 0 wth x the ltx ) lm 0 x l x t u lx l x lm u 0. x l x ) lm sup l x lm f x x l l 3) lm xl x x. lm x l x 0 0. x Defto 3WU Q Y. 005): Suppose that lm EX X 0 p the a sequece of radom varables X p Covergece where Lp X ; X s r.v. ad E X L p X ; X s radom varables of L p f s advaced at a p-th average of for 0 p. I other words X X. I ths paper c s usually sad to be dfferet real costats ad Lemma Lehma E L. 966) Assume that radom varables X ad Y are QD lx s the slowly varyg fucto. The ) EXY EXEY ) PX xy y) PX xpy ) y) x y R 3)If the fuctos of r ad s wth o descedgo cremetal)the r X ) ad sy are QD. Lemma : Assume that ga ) s the fucto of Jot Dstrbuto a 0 ) of X X X that satsfed: a a a f a ) f a m) f a m) m a 0 ) a ) ) 0 a E X f a a ) If there exsts the slowly varyg fucto lx Such that lt ) l lt fort 0 we ca easly get the followg results such that a lt Emax ) [ ] ) a X f a a l t s eve) Proof. Because of radomcty of a usg a fxed. If t s eve ether let t=we have Mathematc Iducto f = we get E X ) f a) Assume that there exsts eve. the two codtos of ad are satsfed by s Set M a a max X. a a If the 94 ISS E-ISS 93-85

3 Moder Appled Scece Vol. 4 o. ; December 00 a X M a a If the a a a X X X a a a a a a a X X X X a a a a a a a a a M X M M Therefore a a a a a a M M X M M Applyg o both sdes of expectatos by equato of Cauchy-Schwarz. a a L a a L L L EM f a ) E X M ) f a ) ) ) a a a L L L f a ) E X ) E M ) f a ) L) ) L L L f a ) f a ) f a ) f a ) L) L) L) L L L f a ) f a ) f a )) f a ) L) L) L) [ L L ) ][ f a ) f a )] L) L) L L [ ] f a ) L) L) L [ L) L ] ) f a L ) LL ) f a L ) L ) L f a ) ) Therefore f =eve the Publshed by Caada Ceter of Scece ad Educato 95

4 Moder Appled Scece Vol. 4 o. ; December 00 a L Emax ) [ ] ) a X f a a L) or codtos of ad are satsfed by s eve. Cosequece the cocluso s satsfed by mathematcal Iducto where eve t a0. Ifer. Let ; X are parwse QD radom sequeces EX 0 EX T ) 0 The Emax T )) ) cl EX. Proof. Sce ) holds E T )) ) r r E T EX EX EX EX EX f ) ad f ) f m) f m) mholds by lemma we have lt Emax T )) ) EX cl EX lt ). Ma results ad the proofs. Theorem : Let f a ) ad ga ) are the fuctos of Jot Dstrbuto a 0 ) of X a Xa Xa that satsfed ad f a ) s satsfed wth supposed that)ad). Assume: ga ) ga m ) ga m) ma 0 3) ga ) c a 0 4) There exst that lx s the slowly varyg fuctosuch that f ) ga ) c l 5) l ) 6) Therefore S s Almost Sure Covergece. ote f rvs.. the S S... as where S X S X. Proof. From6) we have 0 l ). From)4)5) we get E X f a ga ) c c l a l a a. a ) a ) 0 EX for 96 ISS E-ISS 93-85

5 Moder Appled Scece Vol. 4 o. ; December 00 S ; s a sequece of Cauchy satsfyg Hece rvssatsfed.. that ES E S S) 0 For sub-sequece method we frstly prove that S S... as By Marov equato ES S) P S ) S Accordg for the ecessary codtos of progresso covergece the we prove ES S). L because of completeess of L there exst a Applyg the propertes of Lemma ad 4)5)ad the slowly varyg fucto we have c l The we have ES S) lm sup ES S ) lm sup g ) c lm sup g ) l ES S) c l by the B-C lemma we obta ES S) 0 Therefore S S... as Thus accordg Sub-sequece method we shall prove max S S 0. as.. Hece Emax S S )) Applyg the propertes of Lemma ad 3)4)5)ad the slowly varyg fuctowe have l max) ) E S S f l) l g ) c l) l c g ) lm g ) lm g ) c ) Ths compeletes the proof of Theorem. Publshed by Caada Ceter of Scece ad Educato 97

6 Moder Appled Scece Vol. 4 o. ; December 00 Ifer. Let a sequece of costat The S s a.s. Proof. Sce lemma fxg 0 Frst usg the estmate of Defto : ; 0 satsfes EX Y EX Y ) 0 0 a. 0 a E X ) to chec f g. a a a E X) E X XX ) a a aa a EX E X E X a aa a a a EX E X E X a a a a EX E X E X a a a a EX EX EX ) a a a a EX EX a a a EX ) a a EX ) a 0 a ) ) a 0 f a EX a ) ). ga l EX a 0 It s easy to chec that) ) 3) 4) ad 5) holdusg Theorem we obta the fer. Refereces CHE Pg-ya. 008). Lr Covergece for Parwse QD Radom Varables[J]. Acta Math Sceta. 8A3): Jamso B O Rey S & Prutt W. 965). Covergece of Weghted of depedet radom varables. Z. Wahrsch Verb Gebete. 4): Lehma E L. 966). Some cocep ts of depedece. A Math Statst. 43: Matula P. 99). A ote o the almost sure covergece of sums of egatvely depedet radom varables. Statst. Probab. Lett..pp.5:09-3 WA Cheg-gao. 005). Law of Large umbers ad Complete Covergece for Parwse QD Radom Sequeces. Acta Math Appl Sca 8): WAG Y B Chu Su & Xuguo Lu. 998). Some lmt propertes of parwse QD Radom sequeces Appled Mathematcs 3): WAG Y B YA J G & CAI X Z. 00). O the strog stablty for Jamso type weghted product sums of 98 ISS E-ISS 93-85

7 Moder Appled Scece Vol. 4 o. ; December 00 parwse QD seres wth dfferet dstrbuto. Chese A Math. A: WAG Y B YA J-gao & CHEG Feg-yag etal. 00). O the Strog Stablty for Jamso Type Weghted Product Sums of Par-wse QD Seres wth Dfferet Dstrbuto. Chese A Math: Ser A.6): WU Q Y. 00). Covergece propertes of parwse QD radom sequeces. Acta Math Sca 45 3): WU Q Y. 00). The covergece propertes of parwse QD Radom sequeces. Mathematcs 453): WU Q Y. 005). Probablty lmt theory of mxg radom sequeces. Beg:Scece Press. Chapter -3). Publshed by Caada Ceter of Scece ad Educato 99