Research Article On the Strong Laws for Weighted Sums of ρ -Mixing Random Variables

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1 Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2011, Article ID , 8 pages doi: /2011/ Research Article O the Strog Laws for Weighted Sums of ρ -Mixig Radom Variables Xig-Cai Zhou, 1, 2 Chag-Chu Ta, 3 ad Ji-Gua Li 1 1 Departmet of Mathematics, Southeast Uiversity, Najig , Chia 2 Departmet of Mathematics ad Computer Sciece, Toglig Uiversity, Toglig, Ahui , Chia 3 School of Mathematics, Heifei Uiversity of Techology, Hefei, Ahui , Chia Correspodece should be addressed to Chag-Chu Ta, ccta@ustc.edu.c Received 26 October 2010; Revised 5 Jauary 2011; Accepted 27 Jauary 2011 Academic Editor: Matti K. Vuorie Copyright q 2011 Xig-Cai Zhou et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Complete covergece is studied for liear statistics that are weighted sums of idetically distributed ρ -mixig radom variables uder a suitable momet coditio. The results obtaied geeralize ad complemet some earlier results. A Marcikiewicz-Zygmud-type strog law is also obtaied. 1. Itroductio Suppose that {X ; 1} is a sequece of radom variables ad S is a subset of the atural umber set N. LetF S σ X i ; i S, { ρ sup corr f, g ) } : S T N N, dist S, T, f L 2 F S,g L 2 F T, 1.1 where corr f, g ) Cov { f X i ; i S,g X j ; j T )} [ Var { f Xi ; i S } Var { g X j ; j T )}] 1/ Defiitio 1.1. A radom variable sequece {X ; 1} is said to be a ρ -mixig radom variable sequece if there exists k N such that ρ k < 1.

2 2 Joural of Iequalities ad Applicatios The otio of ρ -mixig seems to be similar to the otio of ρ-mixig, but they are quite differet from each other. May useful results have bee obtaied for ρ -mixig radom variables. For example, Bradley 1 has established the cetral limit theorem, Byrc ad Smoleński 2 ad Yag 3 have obtaied momet iequalities ad the strog law of large umbers, Wu 4, 5, Peligrad ad Gut 6, adga 7 have studied almost sure covergece, Utev ad Peligrad 8 have established imal iequalities ad the ivariace priciple, A ad Yua 9 have cosidered the complete covergece ad Marcikiewicz- Zygmud-type strog law of large umbers, ad Budsaba et al. 10 have proved the rate of covergece ad strog law of large umbers for partial sums of movig average processes based o ρ -mixig radom variables uder some momet coditios. For a sequece {X ; 1} of i.i.d. radom variables, Baum ad Katz 11 proved the followig well-kow complete covergece theorem: suppose that {X ; 1} is a sequece of i.i.d. radom variables. The EX 1 0adE X 1 rp < 1 p<2,r 1 if ad oly if r 2 P X i > 1/p ε < for all ε>0. Hsu ad Robbis 12 ad Erdös 13 proved the case r 2adp 1 of the above theorem. The case r 1adp 1 of the above theorem was proved by Spitzer 14. Aad Yua 9 studied the weighted sums of idetically distributed ρ -mixig sequece ad have the followig results. Theorem B. Let {X ; 1} be a ρ -mixig sequece of idetically distributed radom variables, αp > 1, α>1/2, ad suppose that EX 1 0 for α 1. Assume that {a i ;1 i } is a array of real umbers satisfyig a i p O δ, 0 <δ<1, 1.3 { A k 1 i : a i p > k 1 1} e 1/k. 1.4 If E X 1 p <,the αp 2 j P a i X i >ε α <. 1.5 Theorem C. Let {X ; 1} be a ρ -mixig sequece of idetically distributed radom variables, αp > 1, α>1/2, ad EX 1 0 for α 1. Assume that {a i ; 1 i } is array of real umbers satisfyig 1.3.The 1/p a i X i 0 a.s Recetly, Sug 15 obtaied the followig complete covergece results for weighted sums of idetically distributed NA radom variables.

3 Joural of Iequalities ad Applicatios 3 Theorem D. Let {X, X ; 1} be a sequece of idetically distributed NA radom variables, ad let {a i ;1 i, 1} be a array of costats satisfyig A α lim supa α, <, A α, a i α 1.7 for some 0 <α 2.Letb 1/α log 1/γ for some γ>0. Furthermore, suppose that EX 0 where 1 <α 2. If E X α <, for α>γ, E X α log X <, for α γ, 1.8 E X γ <, for α<γ, the 1 P j a i X i >b ε < ε> We fid that the proof of Theorem C is mistakely based o the fact that 1.5 holds for αp 1. Hece, the Marcikiewicz-Zygmud-type strog laws for ρ -mixig sequece have ot bee established. I this paper, we shall ot oly partially geeralize Theorem D to ρ -mixig case, but also exted Theorem B to the case αp 1. The mai purpose is to establish the Marcikiewicz- Zygmud strog laws for liear statistics of ρ -mixig radom variables uder some suitable coditios. We have the followig results. Theorem 1.2. Let {X, X ; 1} be a sequece of idetically distributed ρ -mixig radom variables, ad let {a i ;1 i, 1} be a array of costats satisfyig A β lim supa β, <, A β, a i β, 1.10 where β α, γ for some 0 <α 2 ad γ>0. Letb 1/α log 1/γ.IfEX 0 for 1 <α 2 ad 1.8 for α / γ, the.9 holds. Remark 1.3. The proof of Theorem D was based o Theorem 1 of Che et al. 16, which gave sufficiet coditios about complete covergece for NA radom variables. So far, it is ot kow whether the result of Che et al. 16 holds for ρ -mixig sequece. Hece, we use differet methods from those of Sug 15. We oly exted the case α / γ of Theorem D to ρ -mixig radom variables. It is still ope questio whether the result of Theorem D about the case α γ holds for ρ -mixig sequece.

4 4 Joural of Iequalities ad Applicatios Theorem 1.4. Uder the coditios of Theorem 1.2, the assumptios EX 0 for 1 <α 2 ad 1.8 for α / γ imply the followig Marcikiewicz-Zygmud strog law: b 1 a i X i 0 a.s Proof of the Mai Result Throughout this paper, the symbol C represets a positive costat though its value may chage from oe appearace to ext. It proves coveiet to defie log x 1, l x, where l x deotes the atural logarithm. To obtai our results, the followig lemmas are eeded. Lemma 2.1 Utev ad Peligrad 8. Suppose N is a positive iteger, 0 r<1, ad q 2. The there exists a positive costat D D N, r, q such that the followig statemet holds. If {X i ; i 1} is a sequece of radom variables such that ρ N r with EX i 0 ad E X i q < for every i 1, the for all 1, ) E S i q 1 i D E X i q EX 2 i ) q/2, 2.1 where S i i j 1 X j. Lemma 2.2. Let X be a radom variable ad {a i ; 1 i, 1} be a array of costats satisfyig 1.10, b 1/α log 1/γ.The CE X α 1 P a i X >b CE X γ for α>γ, for α<γ. 2.2 Proof. If γ>α,by a i γ O ad Lyapouov s iequality, the ) 1 1 α/γ a i α a i γ O Hece, 1.7 is satisfied. From the proof of 2.1 of Sug 15, we obtai easily that the result holds.

5 Joural of Iequalities ad Applicatios 5 Proof of Theorem 1.2. Let X i a i X i I a i X i b. For all ε>0, we have 1 P j ) 1 a i X i >εb P a j X j 1 >b P j X i >εb : I 1 I To obtai 1.9, we eed oly to prove that I 1 < ad I 2 <. By Lemma 2.2,oegets 1 I 1 P a j X j 1 >b P a j X >b <. j 1 j Before the proof of I 2 <, we prove firstly b 1 j Ea i X i I a i X i b 0, as. 2.6 For 0 <α 1, b 1 j Ea i X i I a i X i b b 1 E a i X i I a i X i b b α a i α E X α 2.7 C log ) α/γ E X α 0, as. For 1 <α 2, b 1 j Ea i X i I a i X i b b 1 b 1 j Ea i X i I a i X i >b EX i 0 E a i X i I a i X i >b b α a i α E X α 2.8 C log ) α/γ E X α 0, as. Thus 2.6 holds. So, to prove I 2 <, it is eough to show that 1 I 3 P j X i EX i >εb <, ε >0. 2.9

6 6 Joural of Iequalities ad Applicatios By the Chebyshev iequality ad Lemma 2.1, forq {2,γ}, we have I 3 C 1 b q E j q X i EX i C 1 b q E a i X i q I a i X i b 2.10 [ C 1 b q E a i X i 2 I a i X i b : I 31 I 32. ] q/2 For I 31, we cosider the followig two cases. If α<γ,otethate X γ <. We have γ I 31 C 1 b γ a i γ E X γ C α log ) 1 < If α>γ,otethate X α <. we have I 31 C 1 b α a i α E X α C 1 log ) α/γ < Next, we prove I 32 < i the followig two cases. If α<γ 2orγ<α 2, take q> 2, 2γ/α. NotigthatE X α <, we have [ ] q/2 I 32 C 1 b αq/2 a i α E X α C 1 log ) αq/ 2γ < If γ>2 α or γ 2 >α,oegetse X 2 <. Sice a i α O, it implies 1 i a i α C. Therefore, we have a i k a i α a i k α C k α /α C k/α 2.14

7 Joural of Iequalities ad Applicatios 7 for all k α. Hece, a i 2 O 2/α. Takig q>γ, we have [ ] q/2 I 32 C 1 b q a i 2 C 1 b q q/α C 1 log ) q/γ < Proof of Theorem 1.4. By 1.9, a stadard computatio see page 120 of Baum ad Katz 11 or page 1472 of A ad Yua 9, ad the Borel-Catelli Lemma, we have 1 j 2 i j a ix i 2 /α log 2 ) 1/γ 0a.s. i For ay 1, there exists a iteger i such that 2 i 1 <2 i.so 2 i 1 <2 i j 1 a jx j b 1 j 2 i j a jx j 2 i 1 /α log 2 i 1) 1/γ 2 2/α 1 j 2 i j 1 a jx j 2 /α log 2 ) 1/γ ) i 1 1/γ. i From 2.16 ad 2.17, we have lim b 1 a i X i 0a.s Ackowledgmets The authors thak the Academic Editor ad the reviewers for commets that greatly improved the paper. This work is partially supported by Ahui Provicial Natural Sciece Foudatio o M04, Major Programs Foudatio of Miistry of Educatio of Chia o , Natioal Importat Special Project o Sciece ad Techology 2008ZX , ad Natioal Natural Sciece Foudatio of Chia , , ad Refereces 1 R. C. Bradley, O the spectral desity ad asymptotic ormality of weakly depedet radom fields, Joural of Theoretical Probability, vol. 5, o. 2, pp , W. Bryc ad W. Smoleński, Momet coditios for almost sure covergece of weakly correlated radom variables, Proceedigs of the America Mathematical Society, vol. 119, o. 2, pp , S. C. Yag, Some momet iequalities for partial sums of radom variables ad their applicatio, Chiese Sciece Bulleti, vol. 43, o. 17, pp , Q. Y. Wu, Covergece for weighted sums of ρ mixig radom sequeces, Mathematica Applicata, vol. 15, o. 1, pp. 1 4, 2002 Chiese.

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