# On Complete Convergence for Weighted Sums of Arrays of Dependent Random Variables

Authors: Soo Hak Sung
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

A rate of complete convergence for weighted sums of arrays of rowwise independent random variables was obtained by Sung and Volodin (2011). In this paper, we extend this result to negatively associated and negatively dependent random variables. Similar results for sequences ofφ-mixing and ρ*-mixing random variables are also obtained. Our results improve and generalize the results of Baek et al. (2008), Kuczmaszewska (2009), and Wang et al. (2010).


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## Body

## 1. Introduction

The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [1]. A sequence {Xn,n≥1} of random variables converges completely to the constant θ if(1.1)∑n=1∞P(|Xn-θ|>ϵ)<∞∀ϵ>0.
In view of the Borel-Cantelli lemma, this implies that Xn→θ almost surely. Therefore, the complete convergence is a very important tool in establishing almost sure convergence of summation of random variables as well as weighted sums of random variables. Hsu and Robbins [1] proved that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite. Erdös [2] proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors.Ahmed et al. [3] obtained complete convergence for weighted sums of arrays of rowwise independent Banach-space-valued random elements.We recall that the array{Xni,i≥1,n≥1} of random variables is said to be stochastically dominated by a random variable X if(1.2)P(|Xni|>x)≤CP(|X|>x)∀x>0,∀i≥1,n≥1,
where C is a positive constant.Theorem 1.1 (Ahmed et al. [3]).
Let{Xni,i≥1,n≥1} be an array of rowwise independent random elements which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be an array of constants satisfying
(1.3)sup⁡i≥1|ani|=O(n-γ)forsomeγ>0,(1.4)∑i=1∞|ani|=O(nα)forsomeα<γ.
Suppose that there exists δ>1 such that 1+α/γ<δ≤2. Let β≠-1-α and ν=max{1+(1+α+β)/γ,δ}. If E|X|ν<∞ and ∑i=1∞aniXni→0 in probability, then
(1.5)∑n=1∞nβP(‖∑i=1∞aniXni‖>ϵ)<∞∀ϵ>0.Note that there was a typographical error in Ahmed et al. [3] (the relation δ>0 should be δ>1). If β<-1, then the conclusion of Theorem 1.1 is immediate. Hence, Theorem 1.1 is of interest only for β≥-1.Baek et al. [4] extended Theorem 1.1 to negatively associated random variables.Theorem 1.2 (Baek et al. [4]).
Let{Xni,i≥1,n≥1} be an array of rowwise negatively associated random variables which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be an array of constants satisfying (1.3) and (1.4). Suppose that there exists δ>0 such that 1+α/γ<δ≤2. Let β≥-1 and ν=max{1+(1+α+β)/γ,δ}. If EXni=0, for all i≥1 and n≥1, and
(1.6)E|X|log⁡|X|<∞,for1+α+β=0,E|X|ν<∞,for1+α+β>0,
then
(1.7)∑n=1∞nβP(|∑i=1∞aniXni|>ϵ)<∞∀ϵ>0.Sung and Volodin [5] improved Theorem 1.1 as follows.Theorem 1.3 (Sung and Volodin [5]).
Suppose thatβ≥-1. Let {Xni,i≥1,n≥1} be an array of rowwise independent random elements which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be an array of constants satisfying (1.3) and (1.4). Assume that ∑i=1∞aniXni→0 in probability. If
(1.8)E|X|log⁡|X|<∞,for1+α+β=0,E|X|1+(1+α+β)/γ<∞,for1+α+β>0,
then (1.5) holds.In this paper, we extend Theorem1.3 to negatively associated and negatively dependent random variables. We also obtain similar results for sequences of φ-mixing and ρ*-mixing random variables. Our results improve and generalize the results of Baek et al. [4], Kuczmaszewska [6], and Wang et al. [7].Throughout this paper, the symbolC denotes a positive constant which is not necessarily the same one in each appearance. It proves convenient to define logx=max{1,lnx}, where lnx denotes the natural logarithm.


## 2. Preliminaries

In this section, we present some background materials which will be useful in the proofs of our main results.The following lemma is well known, and its proof is standard.Lemma 2.1.
Let{Xn,n≥1} be a sequence of random variables which are stochastically dominated by a random variable X. For any α>0 and b>0, the following statements hold:(i)
E|Xn|αI(|Xn|≤b)≤C{E|X|αI(|X|≤b)+bαP(|X|>b)},(ii)
E|Xn|αI(|Xn|>b)≤CE|X|αI(|X|>b).Lemma 2.2 (Sung [8]).
LetX be a random variable with E|X|r<∞ for some r>0. For any t>0, the following statements hold:(i)
∑n=1∞n-1-tδE|X|r+δI(|X|≤nt)≤CE|X|r for any δ>0,(ii)
∑n=1∞n-1+tδE|X|r-δI(|X|>nt)≤CE|X|r for any δ>0 such that r-δ>0,(iii)
∑n=1∞n-1+trP(|X|>nt)≤CE|X|r.The Rosenthal-type inequality plays an important role in establishing complete convergence. The Rosenthal-type inequalities for sequences of dependent random variables have been established by many authors.The concept of negatively associated random variables was introduced by Alam and Saxena [9] and carefully studied by Joag-Dev and Proschan [10]. A finite family of random variables {Xi,1≤i≤n} is said to be negatively associated if for every pair of disjoint subsets A and B of {1,2,…,n},(2.1)Cov(f1(Xi,i∈A),f2(Xj,j∈B))≤0,
whenever f1 and f2 are coordinatewise increasing and the covariance exists. An infinite family of random variables is negatively associated if every finite subfamily is negatively associated.The following lemma is a Rosenthal-type inequality for negatively associated random variables.Lemma 2.3 (Shao [11]).
Let{Xn,n≥1} be a sequence of negatively associated random variables with EXn=0 and E|Xn|q<∞ for some q≥2 and all n≥1. Then there exists a constant C>0 depending only on q such that
(2.2)E(max⁡1≤j≤n|∑i=1jXi|q)≤C{∑i=1nE|Xi|q+(∑i=1nEXi2)q/2}.The concept of negatively dependent random variables was given by Lehmann [12]. A finite family of random variables {X1,…,Xn} is said to be negatively dependent (or negatively orthant dependent) if for each n≥2, the following two inequalities hold:(2.3)P(X1≤x1,…,Xn≤xn)≤∏i=1nP(Xi≤xi),P(X1>x1,…,Xn>xn)≤∏i=1nP(Xi>xi),
for all real numbers x1,…,xn. An infinite family of random variables is negatively dependent if every finite subfamily is negatively dependent.Obviously, negative association implies negative dependence, but the converse is not true.The following lemma is a Rosenthal-type inequality for negatively dependent random variables.Lemma 2.4 (Asadian et al. [13]).
Let{Xn,n≥1} be a sequence of negatively dependent random variables with EXn=0 and E|Xn|q<∞ for some q≥2 and all n≥1. Then there exists a constant C>0 depending only on q such that
(2.4)E|∑i=1nXi|q≤C{∑i=1nE|Xi|q+(∑i=1nEXi2)q/2}.For a sequence{Xn,n≥1} of random variables defined on a probability space (Ω,ℱ,P), let ℱnm denote the σ-algebra generated by the random variables Xn,Xn+1,…,Xm. Define the φ-mixing coefficients by(2.5)φ(n)=sup⁡k≥1sup⁡{|P(B∣A)-P(B)|,A∈F1k,P(A)≠0,B∈Fk+n∞}.
The sequence {Xn,n≥1} is called φ-mixing (or ϕ-mixing) if φ(n)→0 as n→∞.For anyS⊂ℕ, let ℱS=σ(Xi,i∈S). Define the ρ*-mixing coefficients by(2.6)ρ*(n)=sup⁡corr(f,g),
where the supremum is taken over all S,T⊂ℕ with dist(S,T)≥n, and all f∈L2(ℱS) and g∈L2(ℱT). The sequence {Xn,n≥1} is called ρ*-mixing (or ρ̃-mixing) if there exists k∈ℕ such that ρ*(k)<1.Note that if{Xn,n≥1} is a sequence of independent random variables, then φ(n)=0 and ρ*(n)=0 for all n≥1.The following lemma is a Rosenthal-type inequality forφ-mixing random variables.Lemma 2.5 (Wang et al. [7]).
Let{Xn,n≥1} be a sequence of φ-mixing random variables with EXn=0 and E|Xn|q<∞ for some q≥2 and all n≥1. Assume that ∑n=1∞φ1/2(n)<∞. Then there exists a constant C>0 depending only on q and φ(·) such that
(2.7)E(max⁡1≤j≤n|∑i=1jXi|q)≤C{∑i=1nE|Xi|q+(∑i=1nEXi2)q/2}.The following lemma is a Rosenthal-type inequality forρ*-mixing random variables.Lemma 2.6 (Utev and Peligrad [14]).
Let{Xn,n≥1} be a sequence of random variables with EXn=0 and E|Xn|q<∞ for some q≥2 and all n≥1. If ρ*(k)<1 for some k, then there exists a constant C>0 depending only on q,k, and ρ*(k) such that
(2.8)E(max⁡1≤j≤n|∑i=1jXi|q)≤C{∑i=1nE|Xi|q+(∑i=1nEXi2)q/2}.


## 3. Main Results

In this section, we extend Theorem1.3 to negatively associated and negatively dependent random variables. We also obtain similar results for sequences of φ-mixing and ρ*-mixing random variables.The following theorem extends Theorem1.3 to negatively associated random variables.Theorem 3.1.
Suppose thatβ≥-1. Let {Xni,i≥1,n≥1} be an array of rowwise negatively associated random variables which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be an array of constants satisfying (1.3) and (1.4). If EXni=0 for all i≥1 and n≥1, and (1.8) holds, then
(3.1)∑n=1∞nβP(sup⁡j≥1|∑i=1janiXni|>ϵ)<∞∀ϵ>0.Proof.
Sinceani=ani+-ani-, we may assume that ani≥0. For i≥1 and n≥1, define
(3.2)Xni′=XniI(|Xni|≤nγ)+nγI(Xni>nγ)-nγI(Xni<-nγ),Xni′′=Xni-Xni′.
Then {Xni′,i≥1,n≥1} is still an array of rowwise negatively associated random variables. Moreover, {aniXni′,i≥1,n≥1} is also an array of rowwise negatively associated random variables. Since EXni=0 for all i≥1 and n≥1, it suffices to show that
(3.3)I1=:∑n=1∞nβP(sup⁡j≥1|∑i=1jani(Xni′-EXni′)|>ϵ)<∞,I2=:∑n=1∞nβP(sup⁡j≥1|∑i=1jani(Xni′′-EXni′′)|>ϵ)<∞.
We will prove (3.3) with three cases.
Case 1 (1+(1+α+β)/γ=1(i.e.,1+α+β=0)).
ForI1, we get by Markov’s inequality, Lemmas 2.1–2.3, (1.3), and (1.4) that
(3.4)I1≤ϵ-2∑n=1∞nβEsup⁡j≥1|∑i=1jani(Xni′-EXni′)|2≤C∑n=1∞nβ∑i=1∞|ani|2E|Xni′|2(byLemma2.3)≤C∑n=1∞nβ∑i=1∞|ani|2{E|X|2I(|X|≤nγ)+n2γP(|X|>nγ)}(byLemma2.1)≤C∑n=1∞nβn-γnα{E|X|2I(|X|≤nγ)+n2γP(|X|>nγ)}(by(1.3)and(1.4))≤CE|X|1+(1+α+β)/γ<∞.
The fifth inequality follows from Lemma 2.2.
ForI2, we get by Markov’s inequality, stochastic domination, and (1.4) that
(3.5)I2≤ϵ-1∑n=1∞nβEsup⁡j≥1|∑i=1jani(Xni′′-EXni′′)|≤2ϵ-1∑n=1∞nβ∑i=1∞|ani|E|Xni′′|≤C∑n=1∞nβ∑i=1∞|ani|E|X|I(|X|>nγ)≤C∑n=1∞nβnαE|X|I(|X|>nγ)=C∑n=1∞n-1∑i=n∞E|X|I(iγ<|X|≤(i+1)γ)=C∑i=1∞E|X|I(iγ<|X|≤(i+1)γ)∑n=1in-1≤CE|X|log⁡|X|<∞.Case 2 (1<1+(1+α+β)/γ<2).
As in Case1, we have that I1≤CE|X|1+(1+α+β)/γ<∞.
Similar toI2 in Case 1, we have that
(3.6)I2≤C∑n=1∞nα+βE|X|I(|X|>nγ)=C∑n=1∞nα+β∑i=n∞E|X|I(iγ<|X|≤(i+1)γ)=C∑i=1∞E|X|I(iγ<|X|≤(i+1)γ)∑n=1inα+β≤CE|X|1+(1+α+β)/γ<∞.Case 3 (1+(1+α+β)/γ≥2).
ForI1, we take t>0 sufficiently large such that (γ-α)(1+(1+α+β)/γ+t)/2>1+β. Then we obtain by Markov’s inequality and Lemma 2.3 that
(3.7)I1≤ϵ-1-(1+α+β)/γ-t∑n=1∞nβEsup⁡j≥1|∑i=1jani(Xni′-EXni′)|1+(1+α+β)/γ+t≤C∑n=1∞nβ∑i=1∞E|aniXni′|1+(1+α+β)/γ+t+C∑n=1∞nβ(∑i=1∞E|aniXni′|2)(1+(1+α+β)/γ+t)/2=:I3+I4.Similar to I1 in Case 1, we obtain that
(3.8)I3≤C∑n=1∞nβn-γ((1+α+β)/γ+t)nα{E|X|1+(1+α+β)/γ+tI(|X|≤nγ)+nγ(1+(1+α+β)/γ+t)P(|X|>nγ)}=C∑n=1∞n-1-γtE|X|1+(1+α+β)/γ+tI(|X|≤nγ)+C∑n=1∞nα+β+γP(|X|>nγ)≤CE|X|1+(1+α+β)/γ<∞.Noting E|Xni′|2≤CE|X|2, we obtain by (1.3) and (1.4) that
(3.9)I4≤C∑n=1∞nβ(CE|X|2∑i=1∞|ani|2)(1+(1+α+β)/γ+t)/2≤C∑n=1∞nβ(CE|X|2nα-γ)(1+(1+α+β)/γ+t)/2<∞,
since (γ-α)(1+(1+α+β)/γ+t)/2-β>1. Hence, I1<∞. As in Case 2, we obtain I2≤CE|X|1+(1+α+β)/γ<∞.Remark 3.2.
The moment condition of Theorem3.1 is weaker than that of Theorem 1.2. Also, the conclusion of Theorem 3.1 implies the conclusion of Theorem 1.2. Hence, Theorem 3.1 improves Theorem 1.2. Moreover, the method of the proof of Theorem 3.1 is simpler than that of the proof of Theorem 1.2.Corollary 3.3.
Let{Xni,i≥1,n≥1} be an array of rowwise negatively associated random variables which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be a Toeplitz array satisfying
(3.10)sup⁡i≥1|ani|=O(n1/t-δ)forsomet>0,δ>0.
If
(3.11)E|X|<∞,for0<t<1,E|X|log⁡|X|<∞,fort=1,E|X|1+(1-1/t)/δ<∞,fort>1,
then
(3.12)∑n=1∞P(max⁡1≤j≤n|∑i=1janiXni|>ϵn1/t)<∞∀ϵ>0.Proof.
For the case0<t<1, the result can be easily proved by
(3.13)∑n=1∞P(max⁡1≤j≤n|∑i=1janiXni|>ϵn1/t)≤ϵ-1∑n=1∞n-1/tEmax⁡1≤j≤n|∑i=1janiXni|≤ϵ-1∑n=1∞n-1/t∑i=1n|ani|E|Xni|≤CE|X|∑n=1∞n-1/t<∞.
For the caset≥1, we let bni=anin-1/t. Observe that
(3.14)sup⁡i≥1|bni|=O(n-δ),∑i=1∞|bni|=O(n-1/t).
By Theorem3.1 with α=-1/t,β=0,γ=δ, and ani replaced by bni, we get that
(3.15)∑n=1∞P(max⁡1≤j≤n|∑i=1jbni(Xni-EXni)|>ϵ)<∞∀ϵ>0.
To complete the proof, we only prove that
(3.16)J=:max⁡1≤j≤n|∑i=1jbniEXni|⟶0,
but J≤∑i=1n|bni|E|Xni|≤CE|X|n-1/t→0 as n→∞.Remark 3.4.
When0<t<1, Corollary 3.3 holds without negative association. Kuczmaszewska [6, Corollary  2.4], proved Corollary 3.3 under the stronger moment condition E|X|1+1/δ<∞.The following theorem extends Theorem1.3 to negatively dependent random variables.Theorem 3.5.
Suppose thatβ≥-1. Let {Xni,i≥1,n≥1} be an array of rowwise negatively dependent random variables which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be an array of constants satisfying (1.3) and (1.4). If EXni=0 for all i≥1 and n≥1, and (1.8) holds, then (1.7) holds.Proof.
The proof is the same as that of Theorem3.1 except that we use Lemma 2.4 instead of Lemma 2.3.If the array{Xni,i≥1,n≥1} in Theorem 3.1 is replaced by the sequence {Xn,n≥1}, then we can extend Theorem 3.1 to φ-mixing and ρ*-mixing random variables.Theorem 3.6.
Suppose thatβ≥-1. Let {Xn,n≥1} be a sequence of φ-mixing random variables which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be an array of constants satisfying (1.3) and (1.4). Assume that ∑n=1∞φ1/2(n)<∞. If EXn=0 for all n≥1, and (1.8) holds, then
(3.17)∑n=1∞nβP(sup⁡j≥1|∑i=1janiXi|>ϵ)<∞∀ϵ>0.Proof.
SinceEXn=0 for all n≥1, it suffices to show that
(3.18)∑n=1∞nβP(sup⁡j≥1|∑i=1jani(XiI(|Xi|≤nγ)-EXiI(|Xi|≤nγ))|>ϵ)<∞,∑n=1∞nβP(sup⁡j≥1|∑i=1jani(XiI(|Xi|>nγ)-EXiI(|Xi|>nγ))|>ϵ)<∞.
The rest of the proof is the same as that of Theorem 3.1 except that we use Lemma 2.5 instead of Lemma 2.3 and it is omitted.Remark 3.7.
Can Theorem3.6 be extended to the array {Xni,i≥1,n≥1} of rowwise φ-mixing random variables? Let {φn(i),i≥1} be the sequence of φ-mixing coefficients for the nth row {Xn1,Xn2,…} of the array {Xni}. When we apply Lemma 2.5 to the nth row, the constant C depends on both q and φn(·). That is, the constant C depends on n. Hence we cannot extend Theorem 3.6 to the array by using the method of the proof of Theorem 3.1.Corollary 3.8.
Let{Xn,n≥1} be a sequence of φ-mixing random variables which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be a Toeplitz array satisfying (3.10). Assume that ∑n=1∞φ1/2(n)<∞. If (3.11) holds, then
(3.19)∑n=1∞P(max⁡1≤j≤n|∑i=1janiXi|>ϵn1/t)<∞∀ϵ>0.Proof.
The proof is the same as that of Corollary3.3 except that we use Theorem 3.6 instead of Theorem 3.1.Remark 3.9.
When0<t<1, Corollary 3.8 holds without φ-mixing. Wang et al. [7, Theorem  2.5] proved Corollary 3.8 under the stronger moment condition E|X|max{2/δ,1+1/δ}<∞.Theorem 3.10.
Suppose thatβ≥-1. Let {Xn,n≥1} be a sequence of ρ*-mixing random variables which are stochastically dominated by a random variable X. Let {ani,i≥1,n≥1} be an array of constants satisfying (1.3) and (1.4). If EXn=0 for all n≥1, and (1.8) holds, then (3.17) holds.Proof.
The proof is the same as that of Theorem3.6 except that we use Lemma 2.6 instead of Lemma 2.5.Remark 3.11.
Likewise in Remark3.7, we also cannot extend Theorem 3.10 to the array by using the method of the proof of Theorem 3.1.



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