# On Complete Convergence for Weighted Sums ofρ*-Mixing Random Variables

Authors: Aiting Shen; Xinghui Wang; Huayan Zhu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We prove the strong law of large numbers for weighted sums∑i=1n‍aniXi, which generalizes and improves the corresponding one for independent and identically distributed random variables and φ-mixing random variables. In addition, we present some results on complete convergence for weighted sums of ρ*-mixing random variables under some suitable conditions, which generalize the corresponding ones for independent random variables.


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## Body

## 1. Introduction

Throughout the paper, we letI(A) be the indicator function of the set A. 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 let ψ(0)=0. an=O(bn) denotes that there exists a positive constant C such that |an/bn|≤C. C denotes a positive constant.Let{Xi,i≥1} be a sequence of independent observations from a population distribution. A common expression for these linear statistics is Tn≐∑i=1naniXi, where  the weights ani are either real constants or random variables independent of Xi. Many authors have studied the strong convergence properties for linear statistics Tn and obtained some interesting results. For the details, one can refer to Bai and Cheng [1], Sung [2], Cai [3], Jing and Liang [4], Zhou et al. [5], Wang et al. [6–8] and Wu and Chen [9], Tang [10], and so forth.Recently, Cai [11] proved the following strong law of large numbers for weighted sums of independent and identically distributed random variables.Theorem A.
Let{X,Xn,n≥1} be a sequence of independent and identically distributed random variables with EX=0, EX2<∞, and E[ϕ(|X|)]<∞. Assume that the inverse function ψ(x) of ϕ(x) satisfies
(1)ψ(n)∑i=1n‍1ψ(i)=O(n).
Let {ani,1≤i≤n,n≥1} be a triangular array of constants such that(i)
max1≤i≤n|ani|=O(1/(ψ(n)));(ii)
∑i=1nani2=O(log-1-αn) for some α>0.Then for any ε>0,
(2)∑n=1∞‍n-1P(max1≤j≤n|∑i=1j‍aniXi|>ε)<∞.Wang et al. [12] generalized the result of Theorem A for independent sequences to the case of φ-mixing sequences under conditions of (1), (i), and (ii). Conditions (1) and (i) in Theorem A look ugly, since the conclusion (2) does not contain any information on ψ(n). The main purpose of the paper is to show (2) for ρ*-mixing random variables without conditions of (1) and (i). So our result generalizes and improves the corresponding one of Cai [11] and Wang et al. [12]. In addition, we will present some results on complete convergence for weighted sums of ρ*-mixing random variables under some suitable conditions, which generalize the corresponding ones for independent random variables.Firstly, let us recall the concept ofρ*-mixing random variables.Let{Xn,n≥1} be a sequence of random variables defined on a fixed probability space (Ω,ℱ,P). Write ℱ𝒮=σ(Xi,i∈S⊂N). Given σ-algebras ℬ,ℛ in ℱ, let
(3)ρ(ℬ,ℛ)=supX∈L2(ℬ),Y∈L2(ℛ)|EXY-EXEY|(VarXVarY)1/2.
Define the ρ*-mixing coefficients by
(4)ρ*(k)=sup{ρ(ℱS,ℱT):finitesubsetsS,T⊂N,ρ*(k)=suchthatdist(S,T)≥k},k≥0.
Obviously, 0≤ρ*(k+1)≤ρ*(k)≤1, and ρ*(0)=1.Definition 1.
A sequence{Xn,n≥1} of random variables is said to be ρ*-mixing if there exists k∈N such that ρ*(k)<1.
It is easily seen thatρ*-mixing (i.e., ρ~-mixing) sequence contains independent sequence as a special case. ρ*-mixing random variables were introduced by Bradley [13] and many applications have been found. ρ~-mixing is similar to ρ-mixing, but both are quite different. Many authors have studied this concept providing interesting results and applications. See, for example, Bradley [13] for the central limit theorem, Bryc and Smoleński [14], Peligrad and Gut [15], and Utev and Peligrad [16] for moment inequalities, Gan [17], Kuczmaszewska [18], Wu and Jiang [19] and Wang et al. [20, 21] for almost sure convergence, Peligrad and Gut [15], Cai [22], Kuczmaszewska [23], Zhu [24], An and Yuan [25], Wang et al. [26], and Sung [27] for complete convergence, Peligrad [28] for invariance principle, Wu and Jiang [29] for strong limit theorems for weighted product sums of ρ*-mixing sequences of random variables, Wu and Jiang [30] for Chover-type laws of the k-iterated logarithm, Wu [31] for strong consistency of estimator in linear model, Wang et al. [32] for complete consistency of the estimator of nonparametric regression models, Wu et al. [33] and Guo and Zhu [34] for complete moment convergence, and so forth. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired. So studying the limit behavior of ρ*-mixing random variables is of interest.
The following concepts of slowly varying function and stochastic domination will be used in this work.Definition 2.
A real-valued functionl(x), positive and measurable on (0,∞), is said to be slowly varying if
(5)limx→∞l(xλ)l(x)=1
for each λ>0.Definition 3.
A sequence{Xn,n≥1} of random variables is said to be stochastically dominated by a random variable X if there exists a positive constant C such that
(6)P(|Xn|>x)≤CP(|X|>x)
for all x≥0 and n≥1.This work is organized as follows. Some important lemmas are presented in Section2. Main results and their proofs are provided in Section 3.


## 2. Preliminaries

In this section, we will present some important lemmas which will be used to prove the main results of the paper. The first one is the Rosenthal type maximal inequality forρ*-mixing random variables, which was obtained by Utev and Peligrad [16].Lemma 4 (cf. Utev and Peligrad [16]).
For a positive integerN≥1 and positive real numbers q≥2 and 0≤r<1, there exists a positive constant D=D(q,N,r) such that if {Xn,n≥1} is a sequence of random variables with ρ*(N)<r, EXi=0, and E|Xi|q<∞ for every i≥1, then for all n≥1,
(7)E(max1≤i≤n|∑j=1i‍Xj|q)≤D{∑i=1n‍E|Xi|q+(∑i=1n‍EXi2)q/2}.The next one is a basic property for stochastic domination. For the details of the proof, one can refer to Wu [35] or Tang [36].Lemma 5.
Let{Xn,n≥1} be a sequence of random variables which is stochastically dominated by a random variable X. For any α>0 and b>0, the following two statements hold:
(8)E|Xn|αI(|Xn|≤b)≤C1[E|X|αI(|X|≤b)+bαP(|X|>b)],E|Xn|αI(|Xn|>b)≤C2E|X|αI(|X|>b),
where C1 and C2 are positive constants. Consequently, E|Xn|α≤CE|X|α, where C is a positive constant.The last one is the basic properties for slowly varying function, which was obtained by Bai and Su [37].Lemma 6 (cf. Bai and Su [37]).
Ifl(x)>0 is a slowly varying function as x→∞, then(i)
limx→∞(l(xu)/l(x))=1 for each u>0;(ii)
limk→∞sup2k≤x<2k+1(l(x)/l(2k))=1;(iii)
limx→∞xδl(x)=∞, limx→∞x-δl(x)=0 for each δ>0;(iv)
c12krl(ε2k)≤∑j=1k2jrl(ε2j)≤c22krl(ε2k) for every r>0, ε>0, positive integer k and some c1>0, c2>0;(v)
c32krl(ε2k)≤∑j=k∞2jrl(ε2j)≤c42krl(ε2k) for every r<0, ε>0, positive integer k and some c3>0, c4>0.


## 3. Main Results and Their Proofs

In this section, we will generalize and improve the result of Theorem A for independent and identically distributed random variables to the case ofρ*-mixing random variables. In addition, we will present some results on complete convergence for weighted sums of ρ*-mixing random variables.Our main results are as follows.Theorem 7.
Let{Xn,n≥1} be a sequence of ρ*-mixing random variables, which is stochastically dominated by a random variable X with EXn=0, and EX2<∞. Let {ani,1≤i≤n,n≥1} be a triangular array of constants such that
(9)∑i=1n‍ani2=O(log-1-αn)forsomeα>0.
Then for any ε>0,
(10)∑n=1∞‍n-1P(max1≤j≤n|∑i=1j‍aniXi|>ε)<∞.Proof.
By Markov’s inequality, Lemmas4 and 5, EX2<∞ and condition (9), we have
(11)∑n=1∞‍n-1P(max1≤j≤n|∑i=1j‍aniXi|>ε)≤C∑n=1∞‍n-1E(max1≤j≤n|∑i=1j‍aniXi|2)≤C∑n=1∞‍n-1∑i=1n‍ani2EXi2≤C∑n=1∞‍n-1∑i=1n‍ani2EX2≤C∑n=1∞‍n-1log-1-αn<∞,
which implies (10). This completes the proof of the theorem.Remark 8.
The key to the proof of Theorem7 is the Rosenthal type maximal inequality for ρ*-mixing sequences (i.e., Lemma 4). Similar to the proof of Theorem 7, we have the following result.Theorem 9.
Let{Xn,n≥1} be a sequence of random variables, which is stochastically dominated by a random variable X with EXn=0 and EX2<∞. Let {ani,1≤i≤n,n≥1} be a triangular array of constants such that ∑i=1nani2=O(log-1-αn) for some α>0. Suppose that there exists a positive constant C such that
(12)E(max1≤j≤n|∑i=1j‍aniXi|2)≤C∑i=1n‍ani2EXi2.
Then (10) holds for any ε>0.If the array of constants{ani,1≤i≤n,n≥1} is replaced by the sequence of constants {an,n≥1}, then we can get the following strong law of large numbers for weighted sums ∑i=1naiXi. The proof is standard, so we omit the details.Theorem 10.
Let{Xn,n≥1} be a sequence of random variables, which is stochastically dominated by a random variable X with EXn=0 and EX2<∞. Let {an,n≥1} be a sequence of constants such that ∑i=1nai2=O(log-1-αn)for some α>0. Suppose that there exists a positive constant C such that
(13)E(max1≤j≤n|∑i=1j‍aiXi|2)≤C∑i=1n‍ai2EXi2.
Then for any ε>0,
(14)∑n=1∞‍n-1P(max1≤j≤n|∑i=1j‍aiXi|>ε)<∞,
and in consequence, ∑i=1n‍aiXi→0    a.s.Remark 11.
There are many sequences of random variables satisfying (12), such as negatively associated (NA, in short) sequence (see Shao [38]), negatively superadditive-dependent (NSD, in short) sequence (see Wang et al. [39]), asymptotically almost negatively associated (AANA, in short) sequence (see Yuan and An [40]), φ-mixing sequence (see Wang et al. [12]), and ρ*-mixing sequence (see Utev and Peligrad [16]). Comparing Theorems 7 and 9 with Theorem A, conditions (1) and (i) in Theorem A can be removed. In addition, the condition “identical distribution” in Theorem A can be weakened by “stochastic domination.” Hence, the results of Theorem 7 and Theorem 9 generalize and improve the corresponding one of Theorem A.
In the following, we will present some results on complete convergence for weighted sums ofρ*-mixing random variables. The main ideas are inspired by Kuczmaszewska [41]. The first one is a very general result of complete convergence for weighted sums of ρ*-mixing random variables, which can be applied to obtain other result’s of complete convergence, such as Baum-Katz type complete convergence and Hsu-Robbins type complete convergence.Theorem 12.
Let{Xn,n≥1} be a sequence of ρ*-mixing random variables and let {ani,n≥1,i≥1} be an array of real numbers. Let {bn,n≥1} be an increasing sequence of positive integers and let {cn,n≥1} be a sequence of positive real numbers. If for some q≥2, 0<t<2 and for any ε>0, the following conditions are satisfied:
(15)∑n=1∞‍cn∑i=1bn‍P(|aniXi|≥εbn1/t)<∞,(16)∑n=1∞‍cnbn-(q/t)∑i=1bn‍|ani|qE|Xi|qI(|aniXi|<εbn1/t)<∞,(17)∑n=1∞‍cnbn-(q/2)[∑i=1bn‍ani2EXi2I(|aniXi|<εbn1/t)]q/2<∞,
then
(18)∑n=1∞‍cnP{∑j=1i‍max1≤i≤bn|∑j=1i‍[∑j=1i‍anjXj-anjEXjI(|anjXj|<εbn1/t)∑j=1i‍]|≥εbn1/t∑j=1i‍}<∞.Proof.
Let(19)Yi(n)=aniXiI(|aniXi|<εbn1/t),Sni′=∑j=1i‍Yj(n),Yi(n)=aniXiI(|aniXi|<εbn1/t)n≥1,i≥1,A=⋂i=1bn‍{Yi(n)=aniXi},B=A-=⋃i=1bn‍{Yi(n)≠aniXi}=⋃i=1bn‍(|aniXi|≥εbn1/t),En={max1≤i≤bn|∑j=1i‍[∑j=1i‍anjXj-anjEXjI(|anjXj|<εbn1/t)∑j=1i‍]|≥εbn1/t∑j=1i‍}.
Therefore
(20)P={max1≤i≤bn|∑j=1i‍[∑j=1i‍anjXnj-anjEXjI(|anjXj|<εbn1/t)∑j=1i‍]|≥εbn1/t∑j=1i‍}=P(En)=P(EnA)+P(EnB)≤P(EnA)+P(B)≤∑i=1bn‍P(|aniXi|≥εbn1/t)+ε-qbn-(q/t)E(max1≤i≤bn|Sni′-ESni′|)q.
Using the Cr-inequality and Jensen’s inequality, we can estimate E|Yi(n)-EYi(n)|q in the following way:
(21)E|Yi(n)-EYi(n)|q≤C|ani|qE|Xi|qI(|aniXi|<εbn1/t).
The desired result (18) follows from (15), (16), (17), (20), (21), and Lemma 4 immediately. The proof is completed.In what follows, we will give some applications for Theorem12.Corollary 13.
Let{Xn,n≥1} be a sequence of ρ*-mixing random variables and let {ani,n≥1,≥1} be an array of real numbers. Let l(x)>0 be a slowly varying function as x→∞, α>(1/2), and αr>1. If for some q≥2 and 0<t<2, the following conditions are satisfied for any ε>0:
(22)∑n=1∞‍nαr-2l(n)∑i=1n‍P(|aniXi|≥εn1/t)<∞,(23)∑n=1∞‍n(αr-2)-(q/t)l(n)∑i=1n‍|ani|qE|Xi|qI(|aniXi|<εn1/t)<∞,(24)∑n=1∞‍n(αr-2)-(q/t)l(n)[∑i=1n‍ani2EXi2I(|aniXi|<εn1/t)]q/2<∞,
then
(25)∑n=1∞‍nαr-2l(n)P{max1≤i≤n|∑j=1i‍[∑j=1i‍anjXj-anjEXjI(|anjXj|<εn1/t)∑j=1i‍]|≥εn1/tmax1≤i≤n|∑j=1i‍}<∞.Proof.
Letcn=nαr-2l(n) and let bn=n. The desired result (25) follows from conditions (22)–(24) and Theorem 12 immediately. The proof is completed.Ifani≡1 for n≥1 and i≥1 in Corollary 13, then we can get the following result for ρ*-mixing random variables.Corollary 14.
Let{Xn,n≥1} be a sequence of mean zero ρ*-mixing random variables, which is stochastically dominated by a random variable X and, let l(x)>0 be a slowly varying function as x→∞. If for some α>(1/2), αr>1 and 0<t<2,
(26)E|X|αrtl(|X|t)<∞,
then for any ε>0,
(27)∑n=1∞‍nαr-2l(n)P(max1≤i≤n|∑j=1i‍Xj|≥εn1/t)<∞.Proof.
The proof is similar to that of Corollary  2.8 in Kuczmaszewska [41]. Take ani=1, n≥1, i≥1, and q>max(2,(2t(αr-1))/2-t). In order to prove (27), it is enough to prove that under, the conditions of Corollary 14, conditions (22) and (24) hold.
In fact, by Lemma6 and similar to the proof of Corollary  2.8 in Kuczmaszewska [41], we can obtain
(28)∑n=1∞‍nαr-2l(n)∑i=1n‍P(|Xi|≥εn1/t)≤C∑n=1∞‍nαr-1l(n)P(|X|≥εn1/t)≤C∑k=1∞‍∑n=2k2k+1‍nαr-1l(n)P(|X|≥εn1/t)≤C∑k=1∞‍(2k)αrl(2k)P[|X|≥ε(2k)1/t]≤CE|X|αrtl(|X|t)<∞,
which implies that (22) holds.
LetF(x) be the distribution function of X. It follows, by Lemmas 5 and 6 and the inequality above, that
(29)∑n=1∞‍n(αr-2)-(q/t)l(n)∑i=1n‍E|Xi|qI(|Xi|<εn1/t)≤C∑n=1∞‍nαr-1l(n)P(|X|≥εn1/t)+C∑n=1∞‍n(αr-1)-(q/t)l(n)E|X|qI(|X|<εn1/t)≤C∑k=1∞‍∑n=2k-12k‍n(αr-1)-(q/t)l(n)E|X|qI(|X|<εn1/t)≤C∑k=1∞‍(2k)αr-(q/t)l(2k)∫0(2k)1/t‍|x|qdF(x)≤CE|X|αrtl(|X|t)<∞,
where the fourth inequality above is followed by the proof of Corollary  2.8 in Kuczmaszewska [41]. This shows that (23) holds.
By Lemma5, Cr-inequality, and Markov’s inequality, we can see that
(30)∑n=1∞‍n(αr-2)-(q/t)l(n)[∑i=1n‍EXi2I(|Xi|<εn1/t)]q/2≤C∑n=1∞‍n(αr-2)-(q/t)+(q/2)l(n)×[ε2n2/tP(|X|≥εn1/t)+EX2I(|X|<εn1/t)]q/2≤C∑n=1∞‍n(αr-2)+(q/2)l(n)[P(|X|≥εn1/t)]q/2+C∑n=1∞‍n(αr-2)-(q/t)+(q/2)l(n)[EX2I(|X|<εn1/t)]q/2≤C∑n=1∞‍n(αr-1)(1-(q/2))-1l(n)[E|X|αrt]q/2+C∑n=1∞‍n(αr-2)-(q/t)+(q/2)l(n)[EX2I(|X|<εn1/t)]q/2≤C+C∑n=1∞‍n(αr-2)-(q/t)+(q/2)l(n)×[EX2I(|X|<εn1/t)]q/2<∞,
where the last inequality is followed by the proof of Corollary  2.8 in Kuczmaszewska [41]. Hence, condition (24) is satisfied.
The proof will be completed if we show that(31)n-(1/t)max1≤i≤n|∑j=1i‍EXjI(|Xj|<εn1/t)|⟶0,asn⟶∞.
Ifαrt<1, then we have by Lemma 5 that
(32)n-(1/t)max1≤i≤n|∑j=1i‍EXjI(|Xj|<εn1/t)|≤n-(1/t)∑j=1n‍E|Xj|I(|Xj|<εn1/t)≤CnP(|X|≥εn1/t)+Cn1-(1/t)E|X|I(|X|<εn1/t)≤CnP(|X|αrtεαrt≥n)+Cn1-αrE|X|αrt⟶0,asn⟶∞.
If αrt≥1, then it follows, by EXj=0 and Lemma 5, that
(33)n-(1/t)max1≤i≤n|∑j=1i‍EXjI(|Xj|<εn1/t)|≤n-(1/t)∑j=1n‍|EXjI(|Xj|<εn1/t)|≤n-(1/t)∑j=1n‍E|Xj|I(|Xj|≥εn1/t)≤Cn1-(1/t)E|X|I(|X|≥εn1/t)≤Cn1-αrE|X|αrt⟶0,asn⟶∞.
This completes the proof of the theorem.Remark 15.
Noting that, for typical slowly varying functionsl(x)=1 and l(x)=logx, we can get the simpler formulas in the above theorems.Corollary 16.
Let1≤p≤2 and let {Xn,n≥1} be a sequence of ρ*-mixing random variables with EXn=0 and E|Xn|p<∞ for n≥1. Let {ani,n≥1,i≥1} be an array of real numbers satisfying the condition
(34)∑i=1n‍|ani|pE|Xi|p=O(nδ),asn⟶∞
for some 0<δ≤(2/q) and q>2. Then for any ε>0 and αp≥1,
(35)∑n=1∞‍nαp-2P(max1≤i≤n|∑j=1i‍anjXj|≥εnα)<∞.Proof.
Takecn=nαp-2, bn=n, and (1/t)=α in Theorem 12. Similar to the proof of Corollary  2.2 in Kuczmaszewska [41], we can see that conditions (15)–(17) in Theorem 12 are satisfied by (34).
Noting thatδ<1 and αp≥1, it follows, by EXn=0 for n≥1 and (34), that
(36)1nαmax1≤i≤n|∑j=1i‍anjEXjI(|anjXj|<εnα)|≤1nα∑j=1n‍|anjEXjI(|anjXj|<εnα)|=1nα∑j=1n‍|anjEXjI(|anjXj|≥εnα)|≤1nαp∑j=1n‍|anj|pE|Xj|p≤Cnδ-αp⟶0,asn⟶∞.
The desired result (35) follows from the statements above and Theorem 12 immediately. The proof is completed.



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