The connection between general moment conditions and the applicability of strong laws of large numbers for a sequence of identically distributed ρ-mixing random variables are obtained. 
The results obtained generalize the result of Petrov (1996, Statist. Probab. Lett. 26, 377-380) to ρ-mixing sequences and NA sequences

G. Cai

H. Wu

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Mathematical Communications 10(2005), 63-69 63On the strong laws of large numbers for ρ-mixingsequences∗Guanghui Cai† and Hang Wu‡Abstract. The connection between general moment conditions andthe applicability of strong laws of large numbers for a sequence of iden-tically distributed ρ-mixing random variables are obtained. The resultsobtained generalize the result of Petrov (1996, Statist. Probab. Lett. 26,377-380) to ρ-mixing sequences and NA sequences.Key words: ρ-mixing sequences, NA sequences, strong laws of largenumbers, moment conditionsAMS subject classifications: 60F15Received October 8, 2004 Accepted April 22, 20051. IntroductionLet {Xn, n ≥ 1} be a sequence of i.i.d. random variables. There are two famoustheorems on the strong laws of large numbers for such a sequence: Kolmogorov’stheorem and the Marcinkiewics-Zygmund theorem (see Loe`ve, 1963; Stout, 1974or Petrov, 1995). Let Sn =∑ni=1 Xi. According to Kolmogorov’s theorem, thereexists a constant b such thatlimn→∞|Sn|n= b a.s.if and only if E|X1| <∞ and b = EX1.By the Marcinkiewics-Zygmund theorem, if 0 < p < 2, then the relationslimn→∞|Sn−nb|n1/p= 0 a.s. and E|X1|p < ∞ are equivalent. Here b = 0 if 0 < p < 1,and b = EX1 if 1 ≤ p < 2.Etemadi (1981) proved that Kolmogorov’s theorem remains true if we replace theindependence condition by the weaker condition of pairwise independence of randomvariables. Martikainen (1992) proved some other strong laws of large numbers forpairwise independence of random variables. Petrov (1996) proved the connection∗Research Supported by National Natural Science Foundation of China†Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310 035, P. R. China,e-mail: cghzju@163.com‡Department of Statistics, Zhejiang Gongshang University, Hangzhou 310 035, P. R. China,e-mail: whwhzju@163.com64 G.Cai and H.Wubetween general moment conditions and the applicability of strong laws of largenumbers for a sequence of identically distributed pairwise independence randomvariables sequences.Petrov (1996) proved the following two theorems.Theorem 1. Let {X,Xi, i ≥ 1} be a sequence of pairwise independence identi-cally distributed random variables.IfSnan→ 0 a.s.then Ef(X1) <∞.Theorem 2. Let {X,Xi, i ≥ 1} be a sequence of pairwise independence identi-cally distributed random variables.IfEf(X1) <∞ and∞∑k=n1ak= O(nan),thenSnan→ 0 a.s.A sequence {Xn, n ≥ 1} is said to be a ρ -mixing sequence, if n→∞, we haveρ(n) = supk≥1,X∈L2(Fk1 ),Y ∈L2(F∞k+n)|cov(X,Y )|/‖X‖2‖Y ‖2 → 0,where Fmn is a σ -field generated by the random variables Xn, Xn+1, · · · , Xm. Here‖X‖p = (E|X |p) 1p . As for the ρ -mixing sequence, there are many results, such asShao (1995) for moment inequalities and strong laws of large numbers.As for negatively associated (NA) random variables, Joag (1983) gave the fol-lowing definition.Definition 1 [Joag, 1983]. A finite family of random variables {Xi, 1 ≤ i ≤ n}is said to be negatively associated (NA) if for every pair of disjoint subsets T1 andT2 of {1, 2, ..., n}, we haveCov(f1(Xi, i ∈ T1), f2(Xj , j ∈ T2)) ≤ 0,whenever f1 and f2 are coordinatewise increasing and the covariance exists. Aninfinite family is negatively associated if every finite subfamily is negatively associ-ated.Recently, some authors focused on the problem of limiting behavior of partialsums of NA sequences. Su et al. (1996) derived some moment inequalities of partialsums and a weak convergence for a strong stationary NA sequence. Lin (1997) setup an invariance principal for NA sequences. Su and Qin (1997) also studied somelimiting results for NA sequences. More recently, Liang (1999, 2000) consideredsome complete convergence for weighted sums of NA sequences. Those results,especially some moment inequality by Wang and Xu (2002), Shao (2000) and YangOn the strong laws of large numbers for ρ-mixing sequences 65(2000), undoubtedly propose an important theory guide for further application ofthe NA sequence.In this paper we will examine the connection between general moment conditionsand the applicability of strong laws of large numbers for a sequence of identicallydistributed ρ-mixing random variables. The results obtained generalize the resultof Petrov (1996, Statist. Probab. Lett. 26, 377–380) to ρ-mixing sequences andNA sequences.Throughout this paper C will represent a positive constant, though its valuemay change from one appearance to the next, and an = O(bn) will mean an ≤ Cbn.2. General moment conditions and the strong laws of largenumbersLet f(x) be an even continuous function that is positive and strictly increasing inthe region x > 0 and satisfying the condition f(x)→∞ as x→∞. We putan = f−1(n), (1)where f−1 is the inverse of f . By the properties of f , we have an ↑ ∞.Theorem 3. Let {X,Xi, i ≥ 1} be a sequence of ρ-mixing identically distributedrandom variables. If∞∑i=0ρ(2i) <∞, and Snan→ 0 a.s. (2)thenEf(X1) <∞. (3)In order to prove our results, we need the following lemmas.Lemma 1 [Shao, 1995]. Let {Xi, i ≥ 1} be a sequence of ρ-mixing randomvariables, let Sn =∑ni=1 Xi, EXi = 0, E|Xi|p < ∞ for some p ≥ 2 and for everyi ≥ 1. Then there exists C = C(p), such thatE max1≤i≤n|Si|p ≤ C{exp(C[log n]∑i=0ρ(2i))(n max1≤i≤nE|Xi|2)p/2+n exp(C[log n]∑i=0ρ2/p(2i))max1≤i≤nE|Xi|p}Lemma 2. If events A1, A2, ..., An satisfy V ar(∑nk=1 IAk) ≤ K∑nk=1 P (Ak),then(1− P (n⋃j=1Aj))2 n∑k=1P (Ak) ≤ KP (n⋃j=1Aj). (4)66 G.Cai and H.WuProof. Let β = 1 − P (⋃nk=1 Ak). If β = 0, then (4) is obvious. If β > 0, byV ar(∑nk=1 IAk) ≤ K∑nk=1 P (Ak), thenn∑j=1P (Aj) =n∑j=1P{Aj⋂(n⋃i=1Ai)}= E{ n∑j=1(IAj − P (Aj))I{*ni=1 Ai}}+n∑j=1P (Aj)(1 − β)≤(V ar(n∑j=1IAj )P (n⋃k=1Ak))1/2+n∑j=1P (Aj)(1 − β)≤{Kn∑j=1P (Aj)P (n⋃k=1Ak)}1/2+n∑j=1P (Aj)(1 − β)≤ K2βP( n⋃k=1Ak)+(β2+ 1− β) n∑j=1P (Aj).Thusn∑j=1P (Aj) ≤ K2βP( n⋃k=1Ak)+(1− β2) n∑j=1P (Aj).So we haven∑j=1P (Aj) ≤ Kβ2P( n⋃k=1Ak).✷Lemma 3. Let {Xi, i ≥ 1} be a sequence of ρ-mixing random variables. Let{an} be a sequence of positive numbers such that an−1/an = O(1) and (2) aresatistied. Then ∞∑n=1P (|Xn| > an) <∞. (5)Proof. We observe that the equalityXnan=Snan− an−1anSn−1an−1,then |Xn|an=∣∣∣Snan− an−1anSn−1an−1∣∣∣ ≤∣∣∣Snan∣∣∣+∣∣∣an−1anSn−1an−1∣∣∣.Using an−1/an = O(1) and (2),|Xn|an→ 0 a.s. (6)So we haveP( |Xn|an> 1, i.o.)= 0.On the strong laws of large numbers for ρ-mixing sequences 67Then we haveP( ∞⋂m=1∞⋃n=m{ |Xn|an> 1})= 0.Let An = {|Xn| > an}.So we haveP( ∞⋂m=1∞⋃n=mAn)= 0.There exists an integer m, thenP( ∞⋃n=mAn)<12.By Lemma 1, for all t, we haveV ar( m+t∑n=mI(|Xn| > an))≤ Cm+t∑n=mP (|X | > an).By Lemma 2, we have∞∑n=mP (An) ≤ CP (∞⋃n=mAn),So we have∞∑n=1P (|Xn| > an) =m−1∑n=1P (An) +∞∑n=mP (An) ≤ m− 1 + 12C <∞.✷Proof of Theorem 3. By Lemma 3, an = f−1(n) and f(x) is an even continu-ous function that is positive and strictly increasing in the region x > 0 and satisfiesthe condition f(x)→∞ as x→∞. Then∞∑n=1P (|Xn| > an) <∞.So ∞∑n=1P (f(Xn) > n) <∞.Thus Ef(Xn) <∞. ✷Corollary 1. Let {X,Xi, i ≥ 1} be a sequence of ρ-mixing identically distributedrandom variables. If∑∞i=0 ρ(2i) <∞ and Sn/n1/p → 0 a.s., then E|X |p <∞.Proof. Let f(x) = |x|p, p > 0, then an = f−1(n) = n1/p. Thus by Theorem 3we have the corollary. ✷Remark 1. Corollary 1 is the Marcinkiewics-Zygmund theorem for the ρ-mixingsequence.68 G.Cai and H.WuLemma 4 [Shao, 2000]. Let {Xi, i ≥ 1} be a sequence of NA random variables,EXi = 0, E|Xi|p < ∞ for some p ≥ 2 and for every i ≥ 1. Then there existsC = C(p), such thatE max1≤k≤n∣∣∣k∑i=1Xi∣∣∣p ≤ C{n∑i=1E|Xi|p + (n∑i=1EX2i )p/2}.Theorem 4. Let {X,Xi, i ≥ 1} be a sequence of NA and identically distributedrandom variables. IfSnan→ 0 a.s. (7)thenEf(X1) <∞. (8)Proof. Using Lemma 4, the proof of Theorem 4 is similar to the proof of The-orem 3. ✷Corollary 2. Let {X,Xi, i ≥ 1} be a sequence of NA and identically distributedrandom variables. If∑∞i=0 ρ(2i) <∞ and Sn/n1/p → 0 a.s., then E|X |p <∞.Proof. Let f(x) = |x|p, p > 0, then an = f−1(n) = n1/p. Thus by Theorem 4,we have Corollary 2. ✷Remark 2. Corollary 2 is the Marcinkiewics-Zygmund theorem for NA se-quence.References[1] N.Etemadi, An elementary proof of the strong laws of large numbers, Z.Wahrscheinlichkeitstheorie und verw. Gebiete 55(1981), 119–122.[2] W.T.Huang, B.Xu, Some maximal inequalities and complete convergencesof negatively associated random sequences, Statist. Probab. Lett. 57(2002),183-191.[3] D.K. Joag, F. Proschan, Negative associated of random variables with ap-plication, Ann. Statist. 11(1983), 286–295.[4] H.Y.Liang, C. Su, Complete convergence for weighted sums of NA sequences,Statist. Probab. Lett. 45(1999), 85-95.[5] H.Y.Liang, Complete convergence for weighted sums of negatively associatedrandom variables, Statist. Probab. Lett. 48(2000), 317-325.[6] Z.Y.Lin, Invariance principle for negatively associated sequence, Chinese Sci-ence Bulletin 42(1997), 238-242 (in Chinese).[7] M.Loe`ve, Probability Theory, Van Nostrand, Princeton, 1963.On the strong laws of large numbers for ρ-mixing sequences 69[8] A. I.Martikainen, A note on the strong laws of large numbers for sums ofpairwise independent random variables, Zapiski Nauch. Sem. Petersburg. Otd.Math. Inst. 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