Research Article Some Strong Limit Theorems for Weighted Product Sums of ρ-mixing Sequences of Random Variables

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1 Hdaw Publshg Corporato Joural of Iequaltes ad Applcatos Volume 2009, Artcle ID , 10 pages do: /2009/ Research Artcle Some Strog Lmt Theorems for Weghted Product Sums of ρ-mxg Sequeces of Radom Varables Quyg Wu ad Yuayg Jag College of Scece, Gul Uversty of Techology, Gul , Cha Correspodece should be addressed to Quyg Wu, Receved 4 Jue 2009; Revsed 28 September 2009; Accepted 18 November 2009 Recommeded by Jewge Dshalalow We study almost sure covergece for ρ-mxg sequeces of radom varables. May of the prevous results are our specal cases. For example, the authors exted ad mprove the correspodg results of Che et al ad Wu ad Jag We exted the classcal Jamso covergece theorem ad the Marckewcz strog law of large umbers for depedet sequeces of radom varables to ρ-mxg sequeces of radom varables wthout ecessarly addg ay extra codtos. Copyrght q 2009 Q. Wu ad Y. Jag. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. 1. Itroducto ad Lemmas Let Ω, F,P be a probablty space. The radom varables we deal wth are all defed o Ω, F,P. Let{X ; 1} be a sequece of radom varables. For each oempty set S N, ad wrte F S σ X, S.Gveσ-algebras B, R F,let ρ B, R sup{ corr X, Y ; X L 2 B, Y L 2 R }, 1.1 where corr X, Y EXY EXEY / Var X Var Y. Defe the ρ-mxg coeffcets by ρ sup { ρ F S, F T ; fte subsets S, T N such that dst S, T }, Obvously 0 ρ 1 ρ 1, 0, ad ρ 0 1 except the trval case where all of the radom varables X are degeerate. Defto 1.1. A radom varables sequece {X ; 1} s sad to be a ρ-mxg radom varables sequece f there exsts k N such that ρ k < 1.

2 2 Joural of Iequaltes ad Applcatos ρ-mxg s smlar to ρ-mxg, but both are qute dfferet. A umber of wrters have studed ρ-mxg radom varables sequeces ad a seres of useful results have bee establshed. We refer to Bradley 1 whch assumes ρ k 0 the cetral lmt theorem, Bryc ad Smoleńsk 2, Golde ad Greewood 3 whch assumes ρ 2k <, ad Yag 4 for momet equaltes ad the strog law of large umbers, Wu 5, 6, Wu ad Jag 7, Pelgrad ad Gut 8, adga 9 for almost sure covergece ad Utev ad Pelgrad 10 for maxmal equaltes ad the varace prcple. Whe these are compared wth the correspodg results of depedet radom varables sequeces, there stll remas much to be desred. Lemma 1.2 see 7, Theorem 1. Let {X ; 1} be a ρ-mxg sequece of radom varables whch satsfes Var X < The 1 X EX coverges almost surely a.s.) ad quadratc mea. Lemma 1.3 see 11, Lemma 2.4. For each postve teger m, letg m deote the set of all vectors r, l : r 1,r 2,...,r m, l 1,l 2,...,l m {0, 1, 2,...,m} m {0, 1, 2,...,m} m such that m j 1 r jl j m. The for each postve teger m, there exsts a fucto A m : G m R such that the followg holds. For ay teger m ad ay choce of real umbers x 1,x 2,...,x, oe has that 1 1 < 2 < < m 1 j m x j r, l G m A m )) r, l 1 j m x r j 1 ) lj Ma Results ad the Proof To state our results, we eed some otos. Throughout ths paper, let {ω ; 1} be a sequece of postve real umbers, ad let W 1 ω, 1, satsfy W,ω W 1 0,. Jamso et al. 12 proved the followg result. Suppose that X 1,X 2,... are..d. radom varables wth EX 1 0. Deote N {k; ω 1 k W k }, that s, the umber of subscrpts k such that ω 1 k W k. IfN O, the 1 ω X /W 0 a.s. Che et al. 13 exteded the Jamso Theorem ad obtaed the followg result. Suppose that X 1,X 2,... are..d. radom varables wth EX 1 0, E X 1 r < for some r 1, 2. If N O r, the 1 ω X /W 0a.s. The ma purpose of ths paper s to study the strog lmt theorems for weghted sums of ρ-mxg radom varables sequeces ad try to obta some ew results. We establsh weghted partal sums ad weghted product sums strog covergece theorems. Our results ths paper exted ad mprove the correspodg results of Che et al. 13,WuadJag 7, the classcal Jamso covergece theorem, ad the Marckewcz strog law of large umbers for depedet sequeces of radom varables to ρ-mxg sequeces of radom varables.

3 Joural of Iequaltes ad Applcatos 3 Theorem 2.1. Let {X ; 1} be a sequece of ρ-mxg radom varables wth EX 0, ad let the followg codtos be satsfed: W 1 ω EX I X b 0,, P X b <, Var X I X <b <, 2.3 where b ω 1 W. The T W 1 ω X 0 a.s Theorem 2.2. Suppose that the assumptos of Theorem 2.1 hold, ad also suppose supe X < The for all m 1, U W m ω j X j 0 a.s < < m 1 j m 2.6 Corollary 2.3. Let {X ; 1} be a sequece of ρ-mxg detcally dstrbuted radom varables. Let for some 1 p<2, N {k; b k } c p 1, ad some costat c>0, 2.7 EX 1 0, E X 1 p <. 2.8 The 2.6 holds. Remark 2.4. Let X 1,X 2,... be..d. radom varables, ad p 1 Corollary 2.3, the Corollary 2.3 s the well-kow Jamso covergece theorem. Thus, our Theorem 2.2 ad Corollary 2.3 geeralze ad mprove the Jamso covergece theorem from the..d. case to ρ-mxg sequece. I addto, by Theorems 1 ad 2 Che et al. 13 are specal stuato of Corollary 2.3.

4 4 Joural of Iequaltes ad Applcatos Theorem 2.5. Let {X ; 1} be a sequece of ρ-mxg radom varables. Let {a ; 1} be a sequece of postve real umbers wth a, ad let the followg codtos be satsfed: a 1 a 2 1 EX 2 I X <a <, P X a <, EX I X <a 0, The for all m 1, a m 1 1 < < m 1 j m X j 0, a.s Corollary 2.6. Let {X ; 1} be a ρ-mxg detcally dstrbuted radom varable sequece, for 0 <p<2, E X 1 p <, ad for 1 p<2, EX 1 0. The for all m 1, m/p X j 0, a.s < < m 1 j m 2.13 I partcular, takg m 1, the above formula s the well-kow Marckewcz strog law of large umbers. Thus, our Theorem 2.5 ad Corollary 2.6 geeralze ad mprove the Marckewcz strog law of large umbers from the..d. case to ρ-mxg sequece. I addto, by Theorem 4 Wu ad Jag 7 s a specal case of Corollary 2.6. Proof of Theorem 2.1. Let X b X I X <b.from 2.2, P X b / X 1 P X b < By the Borel-Catell lemma ad the Toepltz lemma, W 1 ω X X b 0 a.s By EX 0ad 2.1, W 1 ω EX b W 1 ω EX I X b 0,

5 Joural of Iequaltes ad Applcatos 5 By 2.3, 1 Var X b 1 Var X I X <b < Applyg Lemma 1.2, b 1 1 X b EX b a.s coverges. Hece W 1 ω X b EX b 0 a.s from the Kroecker lemma. Combg , 2.4 holds. Ths completes the proof of Theorem 2.1. Proof of Theorem 2.2. By Lemma 1.3, U r, l ) G m A m )) r, l 1 j m 1 ω X W 1 ) rj ) lj, 2.20 where G m deote the set of all vectors r, l : r 1,r 2,...,r m, l 1,l 2,...,l m {0, 1, 2,...,m} m {0, 1, 2,...,m} m such that m j 1 r jl j m, ada m r, l are costats whch do ot deped o, {ω ; 1} ad {X ; 1}. Thus, order to prove 2.6, we oly eed to prove that W r ω r Xr 0 a.s. for 1 r m Whe r 1, by Theorem 2.1, 2.21 holds. Whe 2 r m, we get W r ω r Xr 1 ) r/2 ω 2 X from the elemetary equalty a 1 a p a p 1 ap vald for a 0, p 1 appled wth p r/2, a ω 2 X2. Hece, order to prove 2.21, we oly eed to prove that ω 2 X2 0 a.s

6 6 Joural of Iequaltes ad Applcatos By 2.3,usgLemma 1.2, wegetthat b 1 1 X b EX b a.s coverges. By the Kroecker lemma, 1/W 1 ω X b EX b 0a.s. Thus, 1 ω 2 X b EX b 2 1 W 1 2 ω X b EX b ) 0 a.s., 2.25 that s, 1 ω 2 X2 b 2 ω 2 X b EX b W 2 ω 2 EX b 2 0 a.s By 2.3, 1 b 4 Var X b EX b 1 b 4 1 EX b 2 Var X b Var X b < By Lemma 1.2, we have that 1 X b EX b EX b 2) a.s coverges. By the Kroecker lemma, ω 2 1 X b EX b EX b 2) 0 a.s By ω /W 0, ad the Toepltz lemma, ω 2 W ω ω W 1 W 1 ω ω W 0,. 2.30

7 Joural of Iequaltes ad Applcatos 7 The combg 2.5, weobta 0 ) 2 ω 2 EX b 2 supe X W 2 ω Substtutg 2.29 ad , weget ω 2 X2 b 0 a.s The combg 2.2 ad the Borel-Catell lemma, 2.23 holds. Ths completes the proof of Theorem 2.2. Proof of Corollary 2.3. By Theorem 2.2, we oly eed to verfy ad 2.5. FromX havg detcally dstrbuto, 2.7, 2.8 ad 1 p<2, 2.5 holds automatcally. Sce EX I X <b EX 0, 2.33 by the Toepltz lemma, W 1 ω EX I X <b 0, That s, 2.1 holds. By 2.7, P X >b 1 j 1 j 1<b j P X 1 >j 1 ) ) )) N j N j 1 P k 1 X 1 k j 1 k j k ) )) P k 1 X 1 k N j N j 1 j 1 j p P k 1 X 1 k 2.35 E X 1 p <, That s, 2.2 holds.

8 8 Joural of Iequaltes ad Applcatos Smlarly, 1 EX 2 I X <b 1 EX 2 1 I X 1 <b j 1 j 1<b j EX 2 1 I X 1 <b j 1 j 1<b j j 1 ) 2EX 2 1 I X 1 <j ) 2 ) )) j j 1 N j N j 1 EX 2 1 I k 1 X 1 <k j 1 EX 2 1 I ) 2 ) )) k 1 X 1 <k j 1 N j N j 1 j k j EX 2 1 I ) ) 2 k 1 X 1 <k 1 j 2 N j ) j k j EX 2 1 I ) ) 2 k 1 X 1 <k 1 j 2 j p j k EX 2 1 I k 1 X 1 <k k p 2 < E X 1 p I k 1 X1 <k 2.36 E X 1 p <, That s, 2.3 holds. Ths completes proof of Corollary 2.3. Proof of Theorem 2.5. Smlar to the proof of Theorem 2.2, by Lemma 1.3, order to prove 2.12, we oly eed to prove that a r X r 0 a.s. for r 1, Let X a X I X <a, the a 1 X a X X a a 1 1 X a EX a a 1 EX a Whe r 1, by 2.10 ad 2.11, order to prove a 1 1 X 0a.s., we oly eed to prove that a 1 X a EX a 0 a.s

9 Joural of Iequaltes ad Applcatos 9 By 2.9 ad Lemma 1.2, a 1 1 X a EX a a.s coverges. By the Kroecker lemma, 2.39 holds. Whe r 2, by 2.9 ad the Kroecker lemma, a 2 X 2 a 0 a.s By 2.10 ad the Borel-Catell lemma, a 2 X 2 0 a.s Hece, combg 2.39, 2.37 holds. Ths completes the proof of Theorem 2.5. Proof of Corollary 2.6. Let a 1/p. We ca easy to verfy ByTheorem 2.5, Corollary 2.6 holds. Ackowledgmets The authors are very grateful to the referees ad the edtors for ther valuable commets ad some helpful suggestos that mproved the clarty ad readablty of the paper. Ths work was supported by the Natoal Natural Scece Foudato of Cha , the Support Program of the New Cetury Guagx Cha Te-hudred-thousad Talets Project , ad the Guagx, Cha Scece Foudato Refereces 1 R. C. Bradley, O the spectral desty ad asymptotc ormalty of weakly depedet radom felds, Joural of Theoretcal Probablty, vol. 5, o. 2, pp , W. Bryc ad W. Smoleńsk, Momet codtos for almost sure covergece of weakly correlated radom varables, Proceedgs of the Amerca Mathematcal Socety, vol. 119, o. 2, pp , C. M. Golde ad P. E. Greewood, Varace of set-dexed sums of mxg radom varables ad weak covergece of set-dexed processes, The Aals of Probablty, vol. 14, o. 3, pp , S. C. Yag, Some momet equaltes for partal sums of radom varables ad ther applcato, Chese Scece Bullet, vol. 43, o. 17, pp , Q. Y. Wu, Some covergece propertes for ρ-mxg sequeces, Joural of Egeerg Mathematcs, vol. 18, o. 3, pp , 2001 Chese. 6 Q. Y. Wu, Covergece for weghted sums of ρ-mxg radom sequeces, Mathematca Applcata, vol. 15, o. 1, pp. 1 4, 2002 Chese. 7 Q. Y. Wu ad Y. Y. Jag, Some strog lmt theorems for ρ-mxg sequeces of radom varables, Statstcs & Probablty Letters, vol. 78, o. 8, pp , M. Pelgrad ad A. Gut, Almost-sure results for a class of depedet radom varables, Joural of Theoretcal Probablty, vol. 12, o. 1, pp , 1999.

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