AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 216236 10.1155/2013/216236 216236 Research Article On Strong Convergence for Weighted Sums of a Class of Random Variables Shen Aiting Reich Simeon School of Mathematical Science Anhui University, Hefei 230601 China ahu.edu.cn 2013 26 3 2013 2013 05 02 2013 27 02 2013 2013 Copyright © 2013 Aiting Shen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let {Xn,n1} 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).

1. Introduction

Throughout the paper, let I(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 anCbn. Denote logx=ln max(x,e).

Let {X,Xn,n1} be a sequence of identically distributed random variables and {ani,1in,n1} 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 , Cuzick , Wu , Bai and Cheng , Chen and Gan , Cai , Sung [7, 8], Shen , Wang et al. , Zhou et al. , Wu , Xu and Tang , 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  proved the strong law of large numbers for weighted sums: (1)1bni=1naniXi0,a.s., when {X,Xn,n1} 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,1in,n1} is an array of constants satisfying (2)AαlimsupnAα,n<,Aα,nα1ni=1n|ani|α, for some 1<α<2, where bn=n1/α(logn)1/γ+γ(α-1)/α(1+γ).

Cai  generalized the result of Bai and Cheng  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,n1} be a sequence of NA random variables with identical distributions. And let {ani,1in,n1} be a triangular array of constants satisfying i=1n|ani|α=O(n) for 0<α2. Let bn=n1/αlog1/γn for some γ>0. Furthermore, assume that EX=0 when 1<α2. If Eexp(h|X|γ)< for some h>0, then (3)n=11nP(max1jn|i=1janiXi|>ɛbn)<,ɛ>0.

Recently, Wang et al.  extended the result of Cai  for sequences of NA random variables to the case of arrays of rowwise negatively orthant-dependent (NOD, in short) random variables. Sung  improved the result of Cai  for NA random variables under much weaker conditions. Zhou et al.  generalized the result of Sung  to the case of ρ*-mixing random variables when α>γ. The technique used in Sung  is the result of Chen et al.  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 . We not only generalize the result of Zhou et al.  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 αγ. 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 . As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers is obtained.

Theorem 2.

Let {Xn,n1} be a sequence of identically distributed random variables. Let {ani,1in,n1} be an array of constants satisfying i=1n|ani|α=O(n) for some 0<α2. EXn=0 when 1<α2. Let bn=n1/αlog1/γn for some γ>0. Assume that for any q2, there exists a positive constant Cq depending only on q such that (4)E(max1jn|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=1n-1[i=1nP(|aniXi|>bn)]q/2< for Yni=-bnI(aniXi<-bn)+aniXiI(|aniXi|bn)+bnI(aniXi>bn). If (6)E|X1|α<,for  α>γ,E|X1|αlog|X1|<,for  α=γ,E|X1|γ<,for  α<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|αn. It is easy to check that for any ɛ>0, (7)(max1jn|i=1janiXi|>ɛbn)(max1in|aniXi|>bn)(max1jn|i=1jYni|>ɛbn), which implies that (8)P(max1jn|i=1janiXi|>ɛbn)P(max1in|aniXi|>bn)+P(max1jn|i=1jYni|>ɛbn)i=1nP(|aniXi|>bn)+P(max1jn|i=1j(Yni-EYni)|>ɛbn-max1jn|i=1jEYni|). Firstly, we will show that (9)bn-1max1jn|i=1jEYni|0,as  n. When 1<α2, we have by EXn=0, Markov’s inequality and (6) that (10)bn-1max1jn|i=1jEYni|i=1nP(|aniXi|>bn)+bn-1max1jn|i=1janiEXiI(|aniXi|>bn)|i=1nP(|aniX1|>bn)+bn-1i=1nE|aniX1|I(|aniX1|>bn)bn-αi=1n|ani|αE|X1|α+bn-αi=1n|ani|αE|X1|α2E|X1|α(logn)-α/γ0,as  n. When 0<α1, we have by Markov’s inequality and (6) again that (11)bn-1max1jn|i=1jEYni|i=1nP(|aniXi|>bn)+bn-1i=1nE|aniXi|I(|aniXi|bn)=i=1nP(|aniX1|>bn)+bn-1i=1nE|aniX1|I(|aniX1|bn)bn-αi=1n|ani|αE|X1|α+bn-αi=1nE|aniX1|αI(|aniX1|bn)bn-αnE|X1|α+bn-αnE|X1|α=2E|X1|α(logn)-α/γ0,as  n. By (10) and (11), we can get (9) immediately. Hence, for n large enough, (12)P(max1jn|i=1janiXi|>ɛbn)i=1nP(|aniXi|>bn)+P(max1jn|i=1j(Yni-EYni)|>ɛ2bn). To prove (3), we only need to show that (13)In=1n-1i=1nP(|aniXi|>bn)<,(14)Jn=1n-1P(max1jn|i=1j(Yni-EYni)|>ɛ2bn)<. Firstly, we will prove (13). By i=1n|ani|αn and (6), we can get that (15)In=1n-1bn-αi=1nE|aniX1|αI(|aniX1|>bn)n=1n-2(logn)-α/γ×i=1nE|aniX1|αI(i=1n|aniX1|α>n(logn)α/γ)n=1n-2(logn)-α/γ×i=1nE|aniX1|αI(|X1|α>(logn)α/γ)n=1n-1(logn)-α/γE|X1|αI(|X1|γ>logn)=n=1n-1(logn)-α/γ×m=nE|X1|αI(logm<|X1|γlog(m+1))=m=1E|X1|αI(logm<|X1|γlog(m+1))×n=1mn-1(logn)-α/γ{Cm=1E|X1|αI(logm<|X1|γlog(m+1)),forα>γ,Cm=1E|X1|αI(logm<|X1|γlog(m+1))×loglogm,forα=γ,Cm=1E|X1|αI(logm<|X1|γlog(m+1))×(logm)1-α/γ,forα<γ{CE|X1|α,forα>γ,CE|X1|αlog|X1|,for  α=γ,CE|X1|γ,for  α<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)JCn=1n-1bn-qE(max1jn|i=1j(Yni-EYni)|q)Cn=1n-1bn-qi=1nE|Yni|q+Cn=1n-1bn-q(i=1nEYni2)q/2J1+J2. To prove (14), it suffices to show that J1< and J2<.

For j1 and n2, denote (17)Inj={1in:nj+1<|ani|αnj}. In view of i=1n|ani|αn, it is easy to see that {Inj,j1} are disjoint and j=1Inj={1in:ani0}. Hence, we have for all m1 that (18)ni=1n|ani|α={1in:ani0}|ani|α=j=1iInj|ani|αnj=1(j+1)-1Injnj=m(j+1)-q/α(j+1)q/α-1Injnj=m(j+1)-q/α(m+1)q/α-1Inj, which implies that for all m1, (19)j=m(j+1)-q/αInjCm1-q/α,n2.

By Cr’s inequality, (13) and (17), we can get that (20)J1Cn=1n-1i=1nP(|aniXi|>bn)+Cn=1n-1bn-qi=1nE|aniXi|qI(|aniXi|bn)Cn=2n-1bn-qi=1nE|aniX1|qI(|aniX1|bn)Cn=2n-1-q/α(logn)-q/γ×j=1iInjE|aniX1|qI(|X1|(j+1)1/α(logn)1/γ)Cn=2n-1-q/α(logn)-q/γ×j=1nq/αj-q/αE|X1|qI(|X1|(j+1)1/α(logn)1/γ)InjCn=2n-1(logn)-q/γ×j=1j-q/αE|X1|qI(|X1|(logn)1/γ)Inj+Cn=2n-1(logn)-q/γ×j=1j-q/αk=1jE|X1|qI(k1/α(logn)1/γ<|X1|(k+1)1/α(logn)1/γ)InjJ11+J12. If α>γ, we have by (19) and E|X1|α< that (21)J11Cn=2n-1(logn)-q/γE|X1|qI(|X1|(logn)1/γ)Cn=2n-1(logn)-α/γE|X1|αI(|X1|(logn)1/γ)Cn=2n-1(logn)-α/γ<. If αγ, we have by (6) and (19) that (22)J11Cn=2n-1(logn)-q/γE|X1|qI(|X1|(logn)1/γ)Cn=2n-1(logn)-q/γ×m=2nE|X1|qI((log(m-1))1/γ<|X1|(logm)1/γ)=Cm=2E|X1|qI((log(m-1))1/γ<|X1|(logm)1/γ)×n=mn-1(logn)-q/γCm=2(logm)1-q/γE|X1|qI×((log(m-1))1/γ<|X1|(logm)1/γ)Cm=2E|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=Cn=2n-1(logn)-q/γ×k=1E|X1|qI(k1/α(logn)1/γ<|X1|(k+1)1/α(logn)1/γ)×j=kj-q/αInjCn=2n-1(logn)-q/γ×k=1k1-q/αE|X1|qI×(k1/α(logn)1/γ<|X1|(k+1)1/α(logn)1/γ)Cn=2n-1(logn)-α/γ×k=1E|X1|αI(k1/α(logn)1/γ<|X1|(k+1)1/α(logn)1/γ)=Cn=2n-1(logn)-α/γE|X1|αI(|X1|>(logn)1/γ)=Cn=2n-1(logn)-α/γ×m=nE|X1|αI((logm)1/γ<|X1|(log(m+1))1/γ)=Cm=2E|X1|αI((logm)1/γ<|X1|(log(m+1))1/γ)×n=2mn-1(logn)-α/γ{Cm=1E|X1|αI(logm<|X1|γlog(m+1)),forα>γ,Cm=1E|X1|αI(logm<|X1|γlog(m+1))×loglogm,forα=γ,Cm=1E|X1|αI(logm<|X1|γlog(m+1))×(logm)1-α/γ,forα<γ{CE|X1|α,forα>γ,CE|X1|αlog|X1|,for  α=γ,CE|X1|γ,for  α<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)J2Cn=1n-1[i=1nP(|aniXi|>bn)]q/2+Cn=1n-1bn-q[i=1nE|aniXi|2I(|aniXi|bn)]q/2Cn=1n-1[i=1nbn-αE|aniX1|αI(|aniX1|bn)]q/2C(E|X1|α)q/2n=1n-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,n1} be a sequence of identically distributed random variables. Let {an,n1} be a sequence of constants satisfying i=1n|ai|α=O(n) for some 0<α2. EXn=0 when 1<α2. Let bn=n1/αlog1/γn for some γ>0. Assume that for any q2, 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=1n-1[i=1nP(|aiXi|>bn)]q/2< for Yni=-bnI(aiXi<-bn)+aiXiI(|aiXi|bn)+bnI(aiXi>bn). If (6) holds, then (26)n=11nP(max1jn|i=1jaiXi|>ɛbn)<,ɛ>0,(27)1bni=1naiXi0  a.s.,as  n.

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 n1. It follows by (26) that (28)>n=1n-1P(max1jn|Sj|>ɛbn)=i=0n=2i2i+1-1n-1P(max1jn|Sj|>ɛn1/α(logn)1/γ)12i=1P(max1j2i|Sj|>ɛ2(i+1)/α(log2i+1)1/γ). By Borel-Cantelli lemma, we obtain that (29)limimax1j2i|Sj|2(i+1)/α(log2i+1)1/γ=0  a.s. For all positive integers n, there exists a positive integer i0 such that 2i0-1n<2i0. We have by (29) that (30)|Sn|bnmax2i0-1n<2i0|Sn|bn22/αmax1j2i0|Sj|2(i0+1)/α(log2i0+1)1/γ(i0+1i0-1)1/γ0  a.s.,as  i0, 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 Theorem 2. So the details are omitted.

Theorem 4.

Let {Xn,n1} be a sequence of identically distributed random variables. Let {ani,1in,n1} be an array of constants satisfying i=1n|ani|α=O(n) for some 0<α2. EXn=0 when 1<α2. Let bn=n1/αlog1/γn for some γ>0. Assume that for any q2, 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=11nP(|i=1naniXi|>ɛbn)<,ɛ>0.

Remark 5.

There are many sequences of dependent random variables satisfying (4) for all q2. Examples include sequences of NA random variables (see Shao ), ρ*-mixing random variables (see Utev and Peligrad ), φ-mixing random variables with the mixing coefficients satisfying certain conditions (see Wang et al. ), ρ--mixing random variables with the mixing coefficients satisfying certain conditions (see Wang and Lu ), and asymptotically almost negatively associated random variables (see Yuan and An ). There are also many sequences of dependent random variables satisfying (31) for all q2. Examples not only include the sequences of above, but also include sequences of NOD random variables (see Asadian et al. ) and extended negatively dependent random variables (see Shen ).

Acknowledgments

The authors are most grateful to the Editor Simeon Reich and an anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work is supported by the National Natural Science Foundation of China (11201001, 11171001, and 11126176), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20093401120001), the Natural Science Foundation of Anhui Province (1308085QA03, 11040606M12, 1208085QA03), the 211 project of Anhui University, the Youth Science Research Fund of Anhui University, and the Students Science Research Training Program of Anhui University (KYXL2012007).

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