# On Strong Convergence for Weighted Sums of a Class of Random Variables

Authors: Aiting Shen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Let{Xn,n≥1} be a sequence of random variables satisfying the Rosenthal-type maximal inequality. Complete convergence is studied for linear statistics that are weighted sums of identically distributed random variables under a suitable moment condition. As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers is obtained. Our result generalizes the corresponding one of Zhou et al. (2011) and improves the corresponding one of Wang et al. (2011, 2012).


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## Body

## 1. Introduction

Throughout the paper, letI(A) be the indicator function of the set A. C denotes a positive constant which may be different in various places, and an=O(bn) stands for an≤Cbn. Denote logx=ln max(x,e).Let{X,Xn,n≥1} be a sequence of identically distributed random variables and {ani,1≤i≤n,n≥1} an array of constants. The strong convergence results for weighted sums ∑i=1n‍aniXi have been studied by many authors; see, for example, Choi and Sung [1], Cuzick [2], Wu [3], Bai and Cheng [4], Chen and Gan [5], Cai [6], Sung [7, 8], Shen [9], Wang et al. [10–14], Zhou et al. [15], Wu [16–18], Xu and Tang [19], and so forth. Many useful linear statistics are these weighted sums. Examples include least squares estimators, nonparametric regression function estimators, and jackknife estimates among others. Bai and Cheng [4] proved the strong law of large numbers for weighted sums:
(1)1bn∑i=1n‍aniXi→0,a.s.,
when {X,Xn,n≥1} is a sequence of independent and identically distributed random variables with EX=0 and Eexp(h|X|γ)<∞ for some h>0 and γ>0, and {ani,1≤i≤n,n≥1} is an array of constants satisfying
(2)Aα≐limsupn→∞Aα,n<∞,Aα,nα≐1n∑i=1n‍|ani|α,
for some 1<α<2, where bn=n1/α(logn)1/γ+γ(α-1)/α(1+γ).Cai [6] generalized the result of Bai and Cheng [4] to the case of negatively associated (NA, in short) random variables and obtained the following complete convergence result for weighted sums of identically distributed NA random variables.Theorem 1.
Let{X,Xn,n≥1} be a sequence of NA random variables with identical distributions. And let {ani,1≤i≤n,n≥1} be a triangular array of constants satisfying ∑i=1n‍|ani|α=O(n) for 0<α≤2. Let bn=n1/αlog1/γn for some γ>0. Furthermore, assume that EX=0 when 1<α≤2. If Eexp(h|X|γ)<∞ for some h>0, then
(3)∑n=1∞‍1nP(max1≤j≤n|∑i=1j‍aniXi|>ɛbn)<∞,∀ɛ>0.Recently, Wang et al. [14] extended the result of Cai [6] for sequences of NA random variables to the case of arrays of rowwise negatively orthant-dependent (NOD, in short) random variables. Sung [8] improved the result of Cai [6] for NA random variables under much weaker conditions. Zhou et al. [15] generalized the result of Sung [8] to the case of ρ*-mixing random variables when α>γ. The technique used in Sung [8] is the result of Chen et al. [20] for NA random variables, which is not proved for ρ*-mixing random variables. The main purpose of the paper is to further study the strong convergence for a class of random variables satisfying the Rosenthal-type maximal inequality by using a different method from that of Sung [8]. We not only generalize the result of Zhou et al. [15] for ρ*-mixing random variables to the case of a sequence of random variables satisfying the Rosenthal-type maximal inequality, but also consider the case of α≤γ. In addition, our main result improves the corresponding one of Wang et al. [11, 14], since the exponential moment condition is weakened to moment condition.


## 2. Main Results

In this section, we will study the strong convergence for a class of random variables satisfying the Rosenthal-type maximal inequality by using a different method from that of  Sung [8]. As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers is obtained.Theorem 2.
Let{Xn,n≥1} be a sequence of identically distributed random variables. Let {ani,1≤i≤n,n≥1} be an array of constants satisfying ∑i=1n‍|ani|α=O(n) for some 0<α≤2. EXn=0 when 1<α≤2. Let bn=n1/αlog1/γn for some γ>0. Assume that for any q≥2, there exists a positive constant Cq depending only on q such that
(4)E(max1≤j≤n|∑i=1j‍(Yni-EYni)|q)≤Cq[∑i=1n‍E|Yni|q+(∑i=1n‍EYni2)q/2],
where Yni=-bnI(aniXi<-bn)+aniXiI(|aniXi|≤bn)+bnI(aniXi>bn) or Yni=aniXiI(|aniXi|≤bn). Furthermore, suppose that
(5)∑n=1∞‍n-1[∑i=1n‍P(|aniXi|>bn)]q/2<∞
for Yni=-bnI(aniXi<-bn)+aniXiI(|aniXi|≤bn)+bnI(aniXi>bn). If
(6)E|X1|α<∞,forα>γ,E|X1|αlog|X1|<∞,forα=γ,E|X1|γ<∞,forα<γ,
then (3) holds.Proof.
We only need to prove that (3) holds for Yni=-bnI(aniXi<-bn)+aniXiI(|aniXi|≤bn)+bnI(aniXi>bn). The proof for Yni=aniXiI(|aniXi|≤bn) is analogous.
Without loss of generality, we may assume that∑i=1n‍|ani|α≤n. It is easy to check that for any ɛ>0,
(7)(max1≤j≤n|∑i=1j‍aniXi|>ɛbn)⊂(max1≤i≤n|aniXi|>bn)⋃‍(max1≤j≤n|∑i=1j‍Yni|>ɛbn),
which implies that
(8)P(max1≤j≤n|∑i=1j‍aniXi|>ɛbn)≤P(max1≤i≤n|aniXi|>bn)+P(max1≤j≤n|∑i=1j‍Yni|>ɛbn)≤∑i=1n‍P(|aniXi|>bn)+P(max1≤j≤n|∑i=1j‍(Yni-EYni)|>ɛbn-max1≤j≤n|∑i=1j‍EYni|).
Firstly, we will show that
(9)bn-1max1≤j≤n|∑i=1j‍EYni|→0,asn→∞.
When 1<α≤2, we have by EXn=0, Markov’s inequality and (6) that
(10)bn-1max1≤j≤n|∑i=1j‍EYni|≤∑i=1n‍P(|aniXi|>bn)+bn-1max1≤j≤n|∑i=1j‍aniEXiI(|aniXi|>bn)|≤∑i=1n‍P(|aniX1|>bn)+bn-1∑i=1n‍E|aniX1|I(|aniX1|>bn)≤bn-α∑i=1n‍|ani|αE|X1|α+bn-α∑i=1n‍|ani|αE|X1|α≤2E|X1|α(logn)-α/γ→0,asn→∞.
When 0<α≤1, we have by Markov’s inequality and (6) again that
(11)bn-1max1≤j≤n|∑i=1j‍EYni|≤∑i=1n‍P(|aniXi|>bn)+bn-1∑i=1n‍E|aniXi|I(|aniXi|≤bn)=∑i=1n‍P(|aniX1|>bn)+bn-1∑i=1n‍E|aniX1|I(|aniX1|≤bn)≤bn-α∑i=1n‍|ani|αE|X1|α+bn-α∑i=1n‍E|aniX1|αI(|aniX1|≤bn)≤bn-αnE|X1|α+bn-αnE|X1|α=2E|X1|α(logn)-α/γ→0,asn→∞.
By (10) and (11), we can get (9) immediately. Hence, for n large enough,
(12)P(max1≤j≤n|∑i=1j‍aniXi|>ɛbn)≤∑i=1n‍P(|aniXi|>bn)+P(max1≤j≤n|∑i=1j‍(Yni-EYni)|>ɛ2bn).
To prove (3), we only need to show that
(13)I≐∑n=1∞‍n-1∑i=1n‍P(|aniXi|>bn)<∞,(14)J≐∑n=1∞‍n-1P(max1≤j≤n|∑i=1j‍(Yni-EYni)|>ɛ2bn)<∞.
Firstly, we will prove (13). By ∑i=1n‍|ani|α≤n and (6), we can get that
(15)I≤∑n=1∞‍n-1bn-α∑i=1n‍E|aniX1|αI(|aniX1|>bn)≤∑n=1∞‍n-2(logn)-α/γ×∑i=1n‍E|aniX1|αI(∑i=1n‍|aniX1|α>n(logn)α/γ)≤∑n=1∞‍n-2(logn)-α/γ×∑i=1n‍E|aniX1|αI(|X1|α>(logn)α/γ)≤∑n=1∞‍n-1(logn)-α/γE|X1|αI(|X1|γ>logn)=∑n=1∞‍n-1(logn)-α/γ×∑m=n∞‍E|X1|αI(logm<|X1|γ≤log(m+1))=∑m=1∞‍E|X1|αI(logm<|X1|γ≤log(m+1))×∑n=1m‍n-1(logn)-α/γ≤{C∑m=1∞‍E|X1|αI(logm<|X1|γ≤log(m+1)),forα>γ,C∑m=1∞‍E|X1|αI(logm<|X1|γ≤log(m+1))×loglogm,forα=γ,C∑m=1∞‍E|X1|αI(logm<|X1|γ≤log(m+1))×(logm)1-α/γ,forα<γ≤{CE|X1|α,forα>γ,CE|X1|αlog|X1|,forα=γ,CE|X1|γ,forα<γ<∞,
which implies (13).
In the following, we will prove (14). Let q>max{2,α,γ,(2γ/α)}. By Markov’s inequality and condition (4), we have
(16)J≤C∑n=1∞‍n-1bn-qE(max1≤j≤n|∑i=1j‍(Yni-EYni)|q)≤C∑n=1∞‍n-1bn-q∑i=1n‍E|Yni|q+C∑n=1∞‍n-1bn-q(∑i=1n‍EYni2)q/2≐J1+J2.
To prove (14), it suffices to show that J1<∞ and J2<∞.
Forj≥1 and n≥2, denote
(17)Inj={1≤i≤n:nj+1<|ani|α≤nj}.
In view of ∑i=1n‍|ani|α≤n, it is easy to see that {Inj,j≥1} are disjoint and ⋃j=1∞‍Inj={1≤i≤n:ani≠0}. Hence, we have for all m≥1 that
(18)n≥∑i=1n‍|ani|α=∑{1≤i≤n:ani≠0}‍|ani|α=∑j=1∞‍∑i∈Inj‍|ani|α≥n∑j=1∞‍(j+1)-1♯Inj≥n∑j=m∞‍(j+1)-q/α(j+1)q/α-1♯Inj≥n∑j=m∞‍(j+1)-q/α(m+1)q/α-1♯Inj,
which implies that for all m≥1,
(19)∑j=m∞‍(j+1)-q/α♯Inj≤Cm1-q/α,n≥2.
ByCr’s inequality, (13) and (17), we can get that
(20)J1≤C∑n=1∞‍n-1∑i=1n‍P(|aniXi|>bn)+C∑n=1∞‍n-1bn-q∑i=1n‍E|aniXi|qI(|aniXi|≤bn)≤C∑n=2∞‍n-1bn-q∑i=1n‍E|aniX1|qI(|aniX1|≤bn)≤C∑n=2∞‍n-1-q/α(logn)-q/γ×∑j=1∞‍∑i∈Inj‍E|aniX1|qI(|X1|≤(j+1)1/α(logn)1/γ)≤C∑n=2∞‍n-1-q/α(logn)-q/γ×∑j=1∞‍nq/αj-q/αE|X1|qI(|X1|≤(j+1)1/α(logn)1/γ)♯Inj≤C∑n=2∞‍n-1(logn)-q/γ×∑j=1∞‍j-q/αE|X1|qI(|X1|≤(logn)1/γ)♯Inj+C∑n=2∞‍n-1(logn)-q/γ×∑j=1∞‍j-q/α∑k=1j‍E|X1|qI(k1/α(logn)1/γ<|X1|≤(k+1)1/α(logn)1/γ)♯Inj≐J11+J12.
If α>γ, we have by (19) and E|X1|α<∞ that
(21)J11≤C∑n=2∞‍n-1(logn)-q/γE|X1|qI(|X1|≤(logn)1/γ)≤C∑n=2∞‍n-1(logn)-α/γE|X1|αI(|X1|≤(logn)1/γ)≤C∑n=2∞‍n-1(logn)-α/γ<∞.
If α≤γ, we have by (6) and (19) that
(22)J11≤C∑n=2∞‍n-1(logn)-q/γE|X1|qI(|X1|≤(logn)1/γ)≤C∑n=2∞‍n-1(logn)-q/γ×∑m=2n‍E|X1|qI((log(m-1))1/γ<|X1|≤(logm)1/γ)=C∑m=2∞‍E|X1|qI((log(m-1))1/γ<|X1|≤(logm)1/γ)×∑n=m∞‍n-1(logn)-q/γ≤C∑m=2∞‍(logm)1-q/γE|X1|qI×((log(m-1))1/γ<|X1|≤(logm)1/γ)≤C∑m=2∞‍E|X1|γI((log(m-1))1/γ<|X1|≤(logm)1/γ)≤CE|X1|γ<∞.
By (21) and (22), we can get that J11<∞. Next, we will prove that J12<∞.
It follows by (6) and (19) again that
(23)J12=C∑n=2∞‍n-1(logn)-q/γ×∑k=1∞‍E|X1|qI(k1/α(logn)1/γ<|X1|≤(k+1)1/α(logn)1/γ)×∑j=k∞‍j-q/α♯Inj≤C∑n=2∞‍n-1(logn)-q/γ×∑k=1∞‍k1-q/αE|X1|qI×(k1/α(logn)1/γ<|X1|≤(k+1)1/α(logn)1/γ)≤C∑n=2∞‍n-1(logn)-α/γ×∑k=1∞‍E|X1|αI(k1/α(logn)1/γ<|X1|≤(k+1)1/α(logn)1/γ)=C∑n=2∞‍n-1(logn)-α/γE|X1|αI(|X1|>(logn)1/γ)=C∑n=2∞‍n-1(logn)-α/γ×∑m=n∞‍E|X1|αI((logm)1/γ<|X1|≤(log(m+1))1/γ)=C∑m=2∞‍E|X1|αI((logm)1/γ<|X1|≤(log(m+1))1/γ)×∑n=2m‍n-1(logn)-α/γ≤{C∑m=1∞‍E|X1|αI(logm<|X1|γ≤log(m+1)),forα>γ,C∑m=1∞‍E|X1|αI(logm<|X1|γ≤log(m+1))×loglogm,forα=γ,C∑m=1∞‍E|X1|αI(logm<|X1|γ≤log(m+1))×(logm)1-α/γ,forα<γ≤{CE|X1|α,forα>γ,CE|X1|αlog|X1|,forα=γ,CE|X1|γ,forα<γ<∞.
By J11<∞ and J12<∞, we can get that J1<∞.
To prove (14), it suffices to show that J2<∞. By Cr’s inequality, conditions (5) and (6), we can get that
(24)J2≤C∑n=1∞‍n-1[∑i=1n‍P(|aniXi|>bn)]q/2+C∑n=1∞‍n-1bn-q[∑i=1n‍E|aniXi|2I(|aniXi|≤bn)]q/2≤C∑n=1∞‍n-1[∑i=1n‍bn-αE|aniX1|αI(|aniX1|≤bn)]q/2≤C(E|X1|α)q/2∑n=1∞‍n-1(logn)-αq/2γ<∞.
Therefore, (14) follows from (16) and J1<∞, J2<∞ immediately. This completes the proof of the theorem.The following result provides the Marcinkiewicz-Zygmund-type strong law of large numbers for weighted sums∑i=1n‍aiXi of a class of random variables satisfying the Rosenthal-type maximal inequality.Theorem 3.
Let{Xn,n≥1} be a sequence of identically distributed random variables. Let {an,n≥1} be a sequence of constants satisfying ∑i=1n‍|ai|α=O(n) for some 0<α≤2. EXn=0 when 1<α≤2. Let bn=n1/αlog1/γn for some γ>0. Assume that for any q≥2, there exists a positive constant Cq depending only on q such that (4) holds, where Yni=-bnI(aiXi<-bn)+aiXiI(|aiXi|≤bn)+bnI(aiXi>bn) or Yni=aiXiI(|aiXi|≤bn). Furthermore, suppose that
(25)∑n=1∞‍n-1[∑i=1n‍P(|aiXi|>bn)]q/2<∞
for Yni=-bnI(aiXi<-bn)+aiXiI(|aiXi|≤bn)+bnI(aiXi>bn). If (6) holds, then
(26)∑n=1∞‍1nP(max1≤j≤n|∑i=1j‍aiXi|>ɛbn)<∞,∀ɛ>0,(27)1bn∑i=1n‍aiXi→0a.s.,asn→∞.Proof.
The proof of (26) is the same as that of Theorem 2. So the details are omitted. It suffices to show (27). Denote Sn=∑i=1n‍aiXi for each n≥1. It follows by (26) that
(28)∞>∑n=1∞‍n-1P(max1≤j≤n|Sj|>ɛbn)=∑i=0∞‍∑n=2i2i+1-1‍n-1P(max1≤j≤n|Sj|>ɛn1/α(logn)1/γ)≥12∑i=1∞‍P(max1≤j≤2i|Sj|>ɛ2(i+1)/α(log2i+1)1/γ).
By Borel-Cantelli lemma, we obtain that
(29)limi→∞max1≤j≤2i|Sj|2(i+1)/α(log2i+1)1/γ=0a.s.
For all positive integers n, there exists a positive integer i0 such that 2i0-1≤n<2i0. We have by (29) that
(30)|Sn|bn≤max2i0-1≤n<2i0|Sn|bn≤22/αmax1≤j≤2i0|Sj|2(i0+1)/α(log2i0+1)1/γ(i0+1i0-1)1/γ→0a.s.,asi0→∞,
which implies (27). This completes the proof of the theorem.If the Rosenthal type inequality for the maximal partial sum is replaced by the partial sum, then we can get the following complete convergence result for a class of random variables. The proof is similar to that of Theorem2. So the details are omitted.Theorem 4.
Let{Xn,n≥1} be a sequence of identically distributed random variables. Let {ani,1≤i≤n,n≥1} be an array of constants satisfying ∑i=1n‍|ani|α=O(n) for some 0<α≤2. EXn=0 when 1<α≤2. Let bn=n1/αlog1/γn for some γ>0. Assume that for any q≥2, there exists a positive constant Cq depending only on q such that
(31)E(|∑i=1n‍(Yni-EYni)|q)≤Cq[∑i=1n‍E|Yni|q+(∑i=1n‍EYni2)q/2],
where Yni=-bnI(aniXi<-bn)+aniXiI(|aniXi|≤bn)+bnI(aniXi>bn) or Yni=aniXiI(|aniXi|≤bn). Furthermore, suppose that (5) holds for Yni=-bnI(aniXi<-bn)+aniXiI(|aniXi|≤bn)+bnI(aniXi>bn). If (6) satisfies, then
(32)∑n=1∞‍1nP(|∑i=1n‍aniXi|>ɛbn)<∞,∀ɛ>0.Remark 5.
There are many sequences of dependent random variables satisfying (4) for all q≥2. Examples include sequences of NA random variables (see Shao [21]), ρ*-mixing random variables (see Utev and Peligrad [22]), φ-mixing random variables with the mixing coefficients satisfying certain conditions (see Wang et al. [23]), ρ--mixing random variables with the mixing coefficients satisfying certain conditions (see Wang and Lu [24]), and asymptotically almost negatively associated random variables (see Yuan and An [25]). There are also many sequences of dependent random variables satisfying (31) for all q≥2. Examples not only include the sequences of above, but also include sequences of NOD random variables (see Asadian et al. [26]) and extended negatively dependent random variables (see Shen [27]).



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