# Convergence Properties for Asymptotically almost Negatively Associated Sequence

Authors: Xuejun Wang; Shuhe Hu; Wenzhi Yang
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We get the strong law of large numbers, strong growth rate, and the integrability of supremum for the partial sums of asymptotically almost negatively associated sequence. In addition, the complete convergence
for weighted sums of asymptotically almost negatively associated sequences is also studied.


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## Body

## 1. Introduction

Definition 1.1.
A finite collection of random variablesX1,X2,…,Xn is said to be negatively associated (NA) if, for every pair of disjoint subsets A1, A2 of {1,2,…,n},(1.1)Cov{f(Xi:i∈A1),g(Xj:j∈A2)}≤0,
whenever f and g are coordinate-wise nondecreasing such that this covariance exists. An infinite sequence {Xn,n≥1} is NA if every finite subcollection is NA.The concept of negative association was introduced by Joag-Dev and Proschan [1] and Block et al. [2]. By inspecting the proof of maximal inequality for the NA random variables in Matuła [3], one also can allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal [4, 5] introduced the following dependence.Definition 1.2.
A sequence{Xn,n≥1} of random variables is called asymptotically almost negatively associated (AANA) if there exists a nonnegative sequence q(n)→0 as n→∞ such that
(1.2)Cov(f(Xn),g(Xn+1,Xn+2,…,Xn+k))≤q(n)[Var(f(Xn))Var(g(Xn+1,Xn+2,…,Xn+k))]1/2
for all n,k≥1 and for all coordinate-wise nondecreasing continuous functions f and g whenever the variances exist.The family of AANA sequence contains NA (in particular, independent) sequences (withq(n)=0, n≥1) and some more sequences of random variables which are not much deviated from being negatively associated. An example of an AANA sequence which is not NA was constructed by Chandra and Ghosal [4].Since the concept of AANA sequence was introduced by Chandra and Ghosal [4], many applications have been found. For example, Chandra and Ghosal [4] derived the Kolmogorov-type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund, Chandra and Ghosal [5] obtained the almost sure convergence of weighted averages, Ko et al. [6] studied the Hájek-Rényi-type inequality, and Wang et al. [7] established the law of the iterated logarithm for product sums. Recently, Yuan and An [8] established some Rosenthal-type inequalities for maximum partial sums of AANA sequence. As applications of these inequalities, they derived some results on Lp convergence, where 1<p<2, and complete convergence. In addition, they estimated the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.The main purpose of the paper is to study the strong law of large numbers, strong growth rate, and the integrability of supremum for AANA sequence. In addition, the complete convergence for weighted sums of AANA sequence is also studied.Throughout the paper, we let{Xn,n≥1} be a sequence of AANA random variables defined on a fixed probability space (Ω,ℱ,P). Denote Sn≐∑i=1nXi. Let X(a)=-aI(X<-a)+XI(|X|≤a)+aI(X>a) for some a>0, and let I(A) be the indicator function of the set A. For p>1, let q≐p/(p-1) be the dual number of p. We assume that ϕ(x) is a positive increasing function on (0,∞) satisfying ϕ(x)↑∞ as x→∞ and ψ(x) is the inverse function of ϕ(x). Since ϕ(x)↑∞, it follows that ψ(x)↑∞. For easy notation, we let ϕ(0)=0 and ψ(0)=0. The an=O(bn) denotes that there exists a positive constant C such that |an/bn|≤C. C denotes a positive constant which may be different in various places. The main results of this paper are dependent on the following lemmas.Lemma 1.3 (cf. Yuan and An [8, Lemma  2.1]).
Let{Xn,n≥1} be a sequence of AANA random variables with mixing coefficients {q(n),n≥1},  and let f1,f2,… be all nondecreasing (or nonincreasing) functions, then {fn(Xn),n≥1} is still a sequence of AANA random variables with mixing coefficients {q(n),n≥1}.Lemma 1.4.
Let1<p≤2, and let {Xn,n≥1} be a sequence of AANA random variables with mixing coefficients {q(n),n≥1} and EXn=0 for each n≥1.  If ∑n=1∞q2(n)<∞,  then there exists a positive constant Cp depending only on p such that
(1.3)E(max1≤i≤n|Si|p)≤Cp∑i=1nE|Xi|p
for all n≥1, where Cp=2p[22-pp+(6p)p(∑n=1∞q2(n))p/q].Proof.
We use the same notations as that in the study by Yuan and An [8]. They proved that
(1.4)E|max1≤i≤nSi|p≤22-pp∑i=1n∥Xi∥pp+(6p)p(∑i=1n-1q2/q(i)∥Xi∥p)p,E|max1≤i≤n(-Si)|p≤22-pp∑i=1n∥Xi∥pp+(6p)p(∑i=1n-1q2/q(i)∥Xi∥p)p,max1≤i≤n|Si|p≤2p-1|max1≤i≤nSi|p+2p-1|max1≤i≤n(-Si)|p.
By (1.4) and Hölder's inequality, we have
(1.5)E(max1≤i≤n|Si|p)≤2p-1E|max1≤i≤nSi|p+2p-1E|max1≤i≤n(-Si)|p≤2p[22-pp∑i=1nE|Xi|p+(6p)p(∑i=1nq2/q(i)∥Xi∥p)p]≤2p[22-pp∑i=1nE|Xi|p+(6p)p(∑i=1nq2(i))p/q∑i=1nE|Xi|p]≤2p[22-pp+(6p)p(∑n=1∞q2(n))p/q]∑i=1nE|Xi|p=Cp∑i=1nE|Xi|p.
This completes the proof of the lemma.We point out that Lemma1.4 has been studied by Yuan and An [8]. But here we give the accurate coefficient Cp. And Lemma 1.4 generalizes and improves the result of Lemma  2.2 in the study by Ko et al. [6].Lemma 1.5 (cf. Fazekas and Klesov [9, Theorem  2.1] and Hu et al. [10, Lemma  1.5]).
Let{Xn,n≥1} be a sequence of random variables. Let b1,b2,… be a nondecreasing unbounded sequence of positive numbers, and let α1,α2,… be nonnegative numbers.  Let r and C be fixed positive numbers. Assume that, for each n≥1,
(1.6)E(max1≤l≤n|Sl|r)≤C∑l=1nαl,(1.7)∑l=1∞αlblr<∞,
then
(1.8)limn→∞Snbn=0a.s.,
and with the growth rate
(1.9)Snbn=O(βnbn)a.s.,
where
(1.10)βn=max1≤k≤nbkvkδ/r,∀0<δ<1,vn=∑k=n∞αkbkr,limn→∞βnbn=0,E(max1≤l≤n|Slbl|r)≤4C∑l=1nαlblr<∞,E(supl≥1|Slbl|r)≤4C∑l=1∞αlblr<∞.
If further one assumes that αn>0 for infinitely many n, then
(1.11)E(supl≥1|Slβl|r)≤4C∑l=1∞αlβlr<∞.Lemma 1.6 (cf. Fazekas and Klesov [9, Corollary  2.1] and Hu [11, Corollary  2.1.1]).
Letb1,b2,… be a nondecreasing unbounded sequence of positive numbers, and let α1,α2,… be nonnegative numbers. Denote Λk=α1+α2+⋯+αk for k≥1. Let r be a fixed positive number satisfying (1.6). If
(1.12)∑l=1∞Λl(1blr-1bl+1r)<∞,(1.13)Λnbnrisbounded,
then (1.8)–(1.11) hold.Lemma 1.7 (cf. Yuan and An [8, Theorem  2.1]).
Let{Xn,n≥1} be a sequence of AANA random variables with EXi=0 for all i≥1 and p∈(3·2k-1,4·2k-1], where integer number k≥1. If ∑n=1∞qq/p(n)<∞, then there exists a positive constant Dp depending only on p such that, for all n≥1,
(1.14)E(max1≤i≤n|Si|p)≤Dp{∑i=1nE|Xi|p+(∑i=1nEXi2)p/2}.Lemma 1.8.
Assume that the inverse functionψ(x) of  ϕ(x) satisfies
(1.15)ψ(n)∑i=1n1ψ(i)=O(n).
If E[ϕ(|X|)]<∞, then ∑n=1∞(1/ψ(n))E|X|I(|X|>ψ(n))<∞.Proof.
Sinceψ(x) is an increasing function of x, we have that
(1.16)∑n=1∞1ψ(n)E|X|I(|X|>ψ(n))=∑n=1∞1ψ(n)∑i=n∞E|X|I(ψ(i)<|X|≤ψ(i+1))=∑i=1∞E|X|I(ψ(i)<|X|≤ψ(i+1))∑n=1i1ψ(n)≤∑i=1∞P(ψ(i)<|X|≤ψ(i+1))ψ(i+1)∑n=1i1ψ(n)≤C∑i=1∞P(ψ(i)<|X|≤ψ(i+1))i≤CE[ϕ(|X|)]<∞.
The proof is complete.


## 2. Strong Law of Large Numbers and Growth Rate for AANA Sequence

Theorem 2.1.
Let{Xn,n≥1} be a sequence of mean zero AANA random variables with ∑n=1∞q2(n)<∞,  and let {bn,n≥1} be a nondecreasing unbounded sequence of positive numbers; 1<p≤2. Assume that
(2.1)∑n=1∞E|Xn|pbnp<∞,
then
(2.2)limn→∞Snbn=0a.s.,
and with the growth rate
(2.3)Snbn=O(βnbn)a.s.,
where
(2.4)βn=max1≤k≤nbkvkδ/2,∀0<δ<1,vn=∑k=n∞αkbkp,limn→∞βnbn=0,αk=CpE|Xk|p,k≥1,CpisdefinedinLemma1.4,E(max1≤l≤n|Slbl|p)≤4∑l=1nαlblp<∞,E(supl≥1|Slbl|p)≤4∑l=1∞αlblp<∞.
If further one assumes that αn>0 for infinitely many n, then
(2.5)E(supl≥1|Slβl|p)≤4∑l=1∞αlβlp<∞.Proof.
By Lemma1.4, we have
(2.6)E(max1≤k≤n|Sk|p)≤Cp∑k=1nE|Xk|p=∑k=1nαk.
It follows from (2.1) that
(2.7)∑n=1∞αnbnp=Cp∑n=1∞E|Xn|pbnp<∞.
Thus, (2.2)–(2.5) follow from (2.6), (2.7), and Lemma 1.5 immediately. We complete the proof of the theorem.Theorem 2.2.
Let{Xn,n≥1} be a sequence of AANA random variables with ∑n=1∞q2(n)<∞,  1≤p<2.  Denote Qn=max1≤k≤nEXk2 for n≥1 and Q0=0. Assume that
(2.8)∑n=1∞Qnn2/p<∞,
then
(2.9)limn→∞1n1/p∑i=1n(Xi-EXi)=0a.s.,
and with the growth rate
(2.10)1n1/p∑i=1n(Xi-EXi)=O(βnn1/p)a.s.,
where
(2.11)βn=max1≤k≤nk1/pvkδ/2,∀0<δ<1,vn=∑k=n∞αkk2/p,limn→∞βnn1/p=0,αk=C2(kQk-(k-1)Qk-1),k≥1,C2isdefinedinLemma1.4,(2.12)E(max1≤l≤n|Sll1/p|2)≤4∑l=1nαll2/p<∞,(2.13)E(supl≥1|Sll1/p|2)≤4∑l=1∞αll2/p<∞.
If further one assumes that αn>0 for infinitely many n, then
(2.14)E(supl≥1|Slβl|2)≤4∑l=1∞αlβl2<∞.
In addition, for any r∈(0,2),
(2.15)E(supl≥1|Sll1/p|r)≤1+4r2-r∑l=1∞αll2/p<∞.Proof.
Assume thatEXn=0, bn=n1/p, and Λn=∑l=1nαl,n≥1. By Lemma 1.4, we can see that
(2.16)E(max1≤k≤n|∑i=1kXi|2)≤C2∑i=1nEXi2≤C2nQn=∑k=1nαk.
It is a simple fact that αk≥0 for all k≥1. It follows from (2.8) that
(2.17)∑l=1∞Λl(1bl2-1bl+12)=C2∑l=1∞lQl(1l2/p-1(l+1)2/p)≤2C2p∑l=1∞Qll2/p<∞.
That is to say that (1.12) holds. By Remark  2.1 in Fazekas and Klesov [9], (1.12) implies (1.13). By Lemma 1.6, we can obtain (2.9)–(2.14) immediately. By (2.13), it follows that
(2.18)E(supl≥1|Sll1/p|r)=∫0∞P(supl≥1|Sll1/p|r>t)dt≤1+∫1∞P(supl≥1|Sll1/p|>t1/r)dt≤1+E(supl≥1|Sll1/p|2)∫1∞t-2/rdt≤1+4r2-r∑l=1∞αll2/p<∞.
The proof is complete.Theorem 2.3.
Letp∈(3·2k-1,4·2k-1], where integer number k≥1, and let {Xn,n≥1} be a sequence of AANA random variables with EXi=0 for all i≥1 and ∑n=1∞qq/p(n)<∞. Let {bn,n≥1} be a nondecreasing unbounded sequence of positive numbers. Assume that
(2.19)∑n=1∞np/2-1bnpE|Xn|p<∞,∑k=1∞E|Xk|p∑n=k+1∞np/2-2bnp<∞,
then (1.8)–(1.11) hold (for C=1), where
(2.20)α1=2DpE|X1|p,αk=2Dp(kp/2-1∑j=1kE|Xj|p-(k-1)p/2-1∑j=1k-1E|Xj|p),k≥2,
and Dp is defined in Lemma 1.7.Proof.
Sincep>2, 0<2/p<1. By Cr's inequality,
(2.21)(∑i=1n|Xi|p)2/p≤∑i=1nXi2,
which implies that
(2.22)∑i=1nE|Xi|p≤E(∑i=1nXi2)p/2.
By Jensen's inequality, we have
(2.23)(∑i=1nEXi2)p/2≤E(∑i=1nXi2)p/2.
By (2.22)-(2.23) and Cr's inequality,
(2.24)∑i=1nE|Xi|p+(∑i=1nEXi2)p/2≤2E(∑i=1nXi2)p/2≤2np/2-1∑i=1nE|Xi|p.
It follows from Lemma 1.7 and (2.24) that
(2.25)E(max1≤i≤n|Si|p)≤2Dpnp/2-1∑i=1nE|Xi|p=∑l=1nαl.
It is a simple fact that
(2.26)0≤αk≤C1(p)(kp/2-1E|Xk|p+kp/2-2∑j=1k-1E|Xj|p),
where C1(p) is a positive number depending only on p and Dp. By (2.19),
(2.27)∑n=1∞αnbnp≤C1(p)(∑n=1∞np/2-1bnpE|Xn|p+∑k=1∞E|Xk|p∑n=k+1∞np/2-2bnp)<∞.
The desired results follow from (2.25)–(2.27) and Lemma 1.5 immediately.


## 3. Complete Convergence for Weighted Sums of  AANA Random Variables

Theorem 3.1.
Let{X,Xn,n≥1} be a sequence of identically distributed AANA random variables with ∑n=1∞q2(n)<∞,  EX=0,  EX2<∞, and E[ϕ(|X|)]<∞.  Assume that the inverse function ψ(x) of ϕ(x) satisfies (1.15). Let {ani,n≥1,i≥1} be a triangular array of positive constants such that
(i)max1≤i≤nani=O(1/ψ(n)),
(ii)∑i=1nani2=O(log-1-αn) for some α>0.
Then, for anyε>0,
(3.1)∑n=1∞n-1P(max1≤j≤n|∑i=1janiXi|>ε)<∞.Proof.
For eachn≥1, denote
(3.2)Xj(n)=-ψ(n)I(Xj<-ψ(n))+XjI(|Xj|≤ψ(n))+ψ(n)I(Xj>ψ(n)),1≤j≤n,Tj(n)=∑i=1j(aniXi(n)-EaniXi(n)),1≤j≤n,A=⋂i=1n(Xi=Xi(n))=⋂i=1n(|Xi|≤ψ(n)),B=A¯=⋃i=1n(Xi≠Xi(n))=⋃i=1n(|Xi|>ψ(n)),En=(max1≤j≤n|∑i=1janiXi|>ɛ).
It is easy to check that
(3.3)∑i=1janiXi=Tj(n)+∑i=1jEaniXi(n)+∑i=1janiXiI(|Xi|>ψ(n))+∑i=1janiψ(n)[I(Xj<-ψ(n))-I(Xj>ψ(n))],En=EnA+EnB=(max1≤j≤n|Tj(n)+∑i=1jEaniXi(n)|>ɛ)+EnB⊂(max1≤j≤n|Tj(n)|>ɛ-max1≤j≤n|∑i=1jEaniXi(n)|)+B.
Therefore,
(3.4)P(En)≤P(max1≤j≤n|Tj(n)|>ε-max1≤j≤n|∑i=1jEaniXi(n)|)+P(B)≤P(max1≤j≤n|Tj(n)|>ɛ-max1≤j≤n|∑i=1jEaniXi(n)|)+∑i=1nP(|Xi|>ψ(n)).
Firstly, we will show that
(3.5)max1≤j≤n|∑i=1jEaniXi(n)|→0,asn→∞.
It follows from Lemma 1.8 and Kronecker's lemma that
(3.6)1ψ(n)∑i=1nE|X|I(|X|>ψ(i))→0,asn→∞.
By EX=0, condition (i), (3.6), and ψ(n)↑∞, we can see that
(3.7)max1≤j≤n|∑i=1jEaniXi(n)|≤∑i=1n|EaniXiI(|Xi|≤ψ(n))|+∑i=1naniψ(n)EI(|Xi|>ψ(n))≤∑i=1naniE|Xi|I(|Xi|>ψ(n))+∑i=1naniE|Xi|I(|Xi|>ψ(n))≤Cψ(n)∑i=1nE|X|I(|X|>ψ(i))→0,asn→∞,
which implies (3.5). By (3.4) and (3.5), we can see that, for sufficiently large n,
(3.8)P(max1≤j≤n|∑i=1janiXi|>ε)≤P(max1≤j≤n|Tj(n)|>ε2)+∑i=1nP(|Xi|>ψ(n)).
To prove (3.1), it suffices to show that
(3.9)∑n=1∞n-1P(max1≤j≤n|Tj(n)|>ε2)<∞,∑n=1∞n-1∑i=1nP(|Xi|>ψ(n))<∞.
By Markov's inequality, Lemma 1.4, Cr inequality, EX2<∞, and condition (ii), we have
(3.10)∑n=1∞n-1P(max1≤j≤n|Tj(n)|>ε2)≤C∑n=1∞n-1E(max1≤j≤n|Tj(n)|2)≤C∑n=1∞n-1∑i=1nE|aniXi(n)|2≤C∑n=1∞n-1∑i=1nani2EX2I(|X|≤ψ(n))+C∑n=1∞n-1∑i=1nani2ψ2(n)EI(|X|>ψ(n))≤C∑n=1∞n-1∑i=1nani2EX2I(|X|≤ψ(n))+C∑n=1∞n-1∑i=1nani2EX2I(|X|>ψ(n))≤C∑n=1∞n-1∑i=1nani2≤C∑n=1∞n-1log-1-αn<∞.
It follows from E[ϕ(|X|)]<∞ that
(3.11)∑n=1∞n-1∑i=1nP(|Xi|>ψ(n))=∑n=1∞P(|X|>ψ(n))=∑n=1∞P(ϕ(|X|)>n)≤CE[ϕ(|X|)]<∞.Theorem 3.2.
Let{Xn,n≥1} be a sequence of AANA random variables, and let {ani,n≥1,i≥1} be an array of positive numbers. Let {bn,n≥1} be an increasing sequence of positive integers, and let {cn,n≥1} be a sequence of positive numbers. If, for some p∈(3·2k-1,4·2k-1], where integer number k≥1,  0<t<2, and for any ε>0, the following conditions are satisfied:
(3.12)∑n=1∞cn∑i=1bnP(|aniXi|≥εbn1/t)<∞,∑n=1∞cnbn-p/t∑i=1bn|ani|pE|Xi|pI(|aniXi|<εbn1/t)<∞,∑n=1∞cnbn-p/t[∑i=1bnani2EXi2I(|aniXi|<εbn1/t)]p/2<∞,
and ∑n=1∞qq/p(n)<∞, then
(3.13)∑n=1∞cnP{max1≤i≤bn|∑j=1i[anjXj-anjEXjI(|anjXj|<εbn1/t)]|≥εbn1/t}<∞.Proof.
Note that if the series∑n=1∞cn is convergent, then (3.13) holds. Therefore, we will consider only such sequences {cn,n≥1} for which the series ∑n=1∞cn is divergent.
Let(3.14)Yi(n)=-εbn1/tI(aniXi<-εbn1/t)+aniXiI(|aniXi|<εbn1/t)+εbn1/tI(aniXi>εbn1/t),Sni′=∑j=1iYj(n),n≥1,i≥1.
Note that
(3.15)P{max1≤i≤bn|∑j=1i[anjXnj-anjEXjI(|anjXj|<εbn1/t)]|≥εbn1/t}≤C∑i=1bnP(|aniXi|≥εbn1/t)+2pε-pbn-p/tE(max1≤i≤bn|Sni′-ESni′|)p.
Using the Cr inequality and Jensen's inequality, we can estimate E|Yi(n)-EYi(n)|p in the following way:
(3.16)E|Yi(n)-EYi(n)|p≤C|ani|pE|Xi|pI(|aniXi|<εbn1/t)+Cbnp/tP(|aniXi|≥εbn1/t).
By (3.15), (3.16), and Lemma 1.7, we can get
(3.17)P{max1≤i≤bn|∑j=1i[anjXj-anjEXjI(|anjXj|<εbn1/t)]|≥εbn1/t}≤C∑i=1bnP(|aniXi|≥εbn1/t)+Cbn-p/t∑i=1bn|ani|pE|Xi|pI(|aniXi|<εbn1/t)+Cbn-p/t[∑i=1bnani2EXi2I(|aniXi|<εbn1/t)]p/2.
Therefore, we can conclude that (3.13) holds by (3.12) and (3.17).Theorem 3.3.
Let1≤r≤2 and let {Xn,n≥1} be a sequence of AANA random variables with EXn=0 and E|Xn|r<∞ for n≥1.  Let {ani,n≥1,i≥1} be an array of real numbers satisfying the condition
(3.18)∑i=1n|ani|rE|Xi|r=O(nδ)asn→∞
and ∑n=1∞qq/p(n)<∞ for some 0<δ≤2/p and p∈(3·2k-1,4·2k-1], where integer number k≥1. Then, for any ε>0 and αr≥1,
(3.19)∑n=1∞nαr-2P(max1≤i≤n|∑j=1ianjXj|≥εnα)<∞.Proof.
Takecn=nαr-2, bn=n, and 1/t=α in Theorem 3.2. By (3.18), we have
(3.20)∑n=1∞cn∑i=1bnP(|aniXi|≥ɛbn1/t)≤C∑n=1∞nαr-2∑i=1n|ani|rE|Xi|rnαr≤C∑n=1∞n-2+δ<∞,∑n=1∞cnbn-p/t∑i=1bn|ani|pE|Xi|pI(|aniXi|<εbn1/t)≤∑n=1∞n-2∑i=1n|ani|rE|Xi|r≤C∑n=1∞n-2+δ<∞,∑n=1∞cnbn-p/t[∑i=1bnani2EXi2I(|aniXi|<εbn1/t)]p/2≤C∑n=1∞nαr-2-(αrp/2)(∑i=1n|ani|rE|Xi|r)p/2≤C∑n=1∞nαr-2-(αrp/2)+(δp/2)≤C∑n=1∞nαr(1-p/2)-1<∞
following from δp/2≤1. By the assumption EXn=0 for n≥1 and (3.18), we get
(3.21)1nαmax1≤i≤n|∑j=1ianjEXjI(|anjXj|<εnα)|≤1nα∑j=1n|anjEXjI(|anjXj|<εnα)|=1nα∑j=1n|anjEXjI(|anjXj|≥εnα)|≤1nαr∑j=1n|anj|rE|Xj|r≤Cnδ-αr→0,asn→∞
following from δ<1 and αr≥1. We get the desired result from Theorem 3.2 immediately. The proof is complete.Theorem 3.4.
Let{Xn,n≥1} be a sequence of AANA random variables satisfying ∑n=1∞q2(n)<∞, and let {ani,n≥1,i≥1} be an array of positive numbers. Let {bn,n≥1} be an increasing sequence of positive integers, and let {cn,n≥1} be a sequence of positive numbers. If, for some 1<p≤2,  0<t<2, and for any ε>0,  the following conditions are satisfied:
(3.22)∑n=1∞cn∑i=1bnP(|aniXi|≥ɛbn1/t)<∞,∑n=1∞cnbn-p/t∑i=1bn|ani|pE|Xi|pI(|aniXi|<εbn1/t)<∞,
then
(3.23)∑n=1∞cnP{max1≤i≤bn|∑j=1i[anjXj-anjEXjI(|anjXj|<εbn1/t)]|≥εbn1/t}<∞.Proof.
The proof is similar to that of Theorem3.2, so we omit it.



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*Source: 218380-2011-02-07.xml*