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

Abstract

Let {𝑋_𝑛,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).

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 𝑎_𝑛=𝑂(𝑏_𝑛) stands for 𝑎_𝑛≤𝐶𝑏_𝑛. Denote log𝑥=lnmax(𝑥,𝑒).

Let {𝑋,𝑋_𝑛,𝑛≥1} be a sequence of identically distributed random variables and {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} an array of constants. The strong convergence results for weighted sums ∑^𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑖 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: 1𝑏_𝑛^𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑖→0,a.s., (1) when {𝑋,𝑋_𝑛,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 {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} is an array of constants satisfying 𝐴_𝛼≐limsup_𝑛→∞𝐴_𝛼,𝑛<∞,𝐴^𝛼_𝛼,𝑛≐1𝑛^𝑛_𝑖=1𝑎_𝑛𝑖^𝛼, (2) for some 1<𝛼<2, where 𝑏_𝑛=𝑛^1/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,𝑋_𝑛,𝑛≥1} be a sequence of NA random variables with identical distributions. And let {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be a triangular array of constants satisfying ∑^𝑛_𝑖=1|𝑎_𝑛𝑖|^𝛼=𝑂(𝑛) for 0<𝛼≤2. Let 𝑏_𝑛=𝑛^1/𝛼log^1/𝛾𝑛 for some 𝛾>0. Furthermore, assume that 𝐸𝑋=0 when 1<𝛼≤2. If 𝐸exp(ℎ|𝑋|^𝛾)<∞ for some ℎ>0, then ^∞_𝑛=11𝑛𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜀𝑏_𝑛<∞,∀𝜀>0. (3)

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 {𝑋_𝑛,𝑛≥1} be a sequence of identically distributed random variables. Let {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be an array of constants satisfying ∑^𝑛_𝑖=1|𝑎_𝑛𝑖|^𝛼=𝑂(𝑛) for some 0<𝛼≤2. 𝐸𝑋_𝑛=0 when 1<𝛼≤2. Let 𝑏_𝑛=𝑛^1/𝛼log^1/𝛾𝑛 for some 𝛾>0. Assume that for any 𝑞≥2, there exists a positive constant 𝐶_𝑞 depending only on 𝑞 such that 𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑌_𝑛𝑖−𝐸𝑌_𝑛𝑖^𝑞≤𝐶_𝑞^𝑛_𝑖=1𝐸𝑌_𝑛𝑖^𝑞+^𝑛_𝑖=1𝐸𝑌²_𝑛𝑖^𝑞/2, (4) where 𝑌_𝑛𝑖=−𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖<−𝑏_𝑛)+𝑎_𝑛𝑖𝑋_𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑖|≤𝑏_𝑛)+𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛) or 𝑌_𝑛𝑖=𝑎_𝑛𝑖𝑋_𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑖|≤𝑏_𝑛). Furthermore, suppose that ^∞_𝑛=1𝑛^−1𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛^𝑞/2<∞ (5) for Y_𝑛𝑖=−𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖<−𝑏_𝑛)+𝑎_𝑛𝑖𝑋_𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑖|≤𝑏_𝑛)+𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛). If 𝐸𝑋₁^𝛼𝐸𝑋<∞,for𝛼>𝛾,_1𝛼log𝑋₁𝐸𝑋<∞,for𝛼=𝛾,_1𝛾<∞,for𝛼<𝛾, (6) then (3) holds.

Proof. We only need to prove that (3) holds for 𝑌_𝑛𝑖=−𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖<−𝑏_𝑛)+𝑎_𝑛𝑖𝑋_𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑖|≤𝑏_𝑛)+𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛). The proof for 𝑌_𝑛𝑖=𝑎_𝑛𝑖𝑋_𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑖|≤𝑏_𝑛) is analogous.
Without loss of generality, we may assume that ∑^𝑛_𝑖=1|𝑎_𝑛𝑖|^𝛼≤𝑛. It is easy to check that for any 𝜀>0, max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜀𝑏_𝑛⊂max_{1≤𝑖≤𝑛}𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑌_𝑛𝑖>𝜀𝑏_𝑛, (7) which implies that 𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜀𝑏_𝑛≤𝑃max_{1≤𝑖≤𝑛}𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛+𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑌_𝑛𝑖>𝜀𝑏_𝑛≤^𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛+𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑌_𝑛𝑖−𝐸𝑌_𝑛𝑖>𝜀𝑏_𝑛−max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝐸𝑌_𝑛𝑖. (8) Firstly, we will show that 𝑏_𝑛⁻¹max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝐸𝑌_𝑛𝑖→0,as𝑛→∞. (9) When 1<𝛼≤2, we have by 𝐸𝑋_𝑛=0, Markov’s inequality and (6) that 𝑏_𝑛⁻¹max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝐸𝑌_𝑛𝑖≤^𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛+𝑏_𝑛⁻¹max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖𝐼𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛≤^𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋₁>𝑏_𝑛+𝑏_𝑛^𝑛−1_𝑖=1𝐸𝑎_𝑛𝑖𝑋₁𝐼𝑎_𝑛𝑖𝑋₁>𝑏_𝑛≤𝑏_𝑛^𝑛−𝛼_𝑖=1𝑎_𝑛𝑖^𝛼𝐸𝑋₁^𝛼+𝑏_𝑛^𝑛−𝛼_𝑖=1𝑎_𝑛𝑖^𝛼𝐸𝑋₁^𝛼≤2𝐸𝑋₁^𝛼log𝑛^−𝛼/𝛾→0,as𝑛→∞. (10) When 0<𝛼≤1, we have by Markov’s inequality and (6) again that 𝑏_𝑛⁻¹max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝐸𝑌_𝑛𝑖≤^𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛+𝑏_𝑛^𝑛−1_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑖𝐼𝑎_𝑛𝑖𝑋_𝑖≤𝑏_𝑛=^𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋₁>𝑏_𝑛+𝑏_𝑛^𝑛−1_𝑖=1𝐸𝑎_𝑛𝑖𝑋₁𝐼𝑎_𝑛𝑖𝑋₁≤𝑏_𝑛≤𝑏_𝑛^𝑛−𝛼_𝑖=1𝑎_𝑛𝑖^𝛼𝐸𝑋₁^𝛼+𝑏_𝑛^𝑛−𝛼_𝑖=1𝐸𝑎_𝑛𝑖𝑋₁^𝛼𝐼𝑎_𝑛𝑖𝑋₁≤𝑏_𝑛≤𝑏_𝑛^−𝛼𝑛𝐸𝑋₁^𝛼+𝑏_𝑛^−𝛼𝑛𝐸𝑋₁^𝛼=2𝐸𝑋₁^𝛼log𝑛^−𝛼/𝛾→0,as𝑛→∞. (11) By (10) and (11), we can get (9) immediately. Hence, for 𝑛 large enough, 𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜀𝑏_𝑛≤^𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛+𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑌_𝑛𝑖−𝐸𝑌_𝑛𝑖>𝜀2𝑏_𝑛. (12) To prove (3), we only need to show that 𝐼≐^∞_𝑛=1𝑛^𝑛−1_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛<∞,𝐽≐^∞_𝑛=1𝑛⁻¹𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑌_𝑛𝑖−𝐸𝑌_𝑛𝑖>𝜀2𝑏_𝑛<∞. (13)(14) Firstly, we will prove (13). By ∑^𝑛_𝑖=1|𝑎_𝑛𝑖|^𝛼≤𝑛 and (6), we can get that 𝐼≤^∞_𝑛=1𝑛⁻¹𝑏_𝑛^𝑛−𝛼_𝑖=1𝐸𝑎_𝑛𝑖𝑋₁^𝛼𝐼𝑎_𝑛𝑖𝑋₁>𝑏_𝑛≤^∞_𝑛=1𝑛⁻²log𝑛^−𝛼/𝛾×^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋₁^𝛼𝐼^𝑛_𝑖=1𝑎_𝑛𝑖𝑋₁^𝛼>𝑛log𝑛^𝛼/𝛾≤^∞_𝑛=1𝑛⁻²log𝑛^−𝛼/𝛾×^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋₁^𝛼𝐼𝑋₁^𝛼>log𝑛^𝛼/𝛾≤^∞_𝑛=1𝑛⁻¹log𝑛^−𝛼/𝛾𝐸𝑋₁^𝛼𝐼𝑋₁^𝛾=>log𝑛_∞𝑛=1𝑛⁻¹log𝑛^−𝛼/𝛾×^∞_𝑚=𝑛𝐸𝑋₁^𝛼𝐼log𝑚<𝑋₁^𝛾()=≤log𝑚+1_∞𝑚=1𝐸𝑋₁^𝛼𝐼log𝑚<𝑋₁^𝛾(×≤log𝑚+1)_𝑚𝑛=1𝑛⁻¹log𝑛^−𝛼/𝛾≤𝐶∑^∞_𝑚=1𝐸𝑋₁^𝛼𝐼log𝑚<𝑋₁^𝛾(),𝐶≤log𝑚+1for𝛼>𝛾,_∞𝑚∑₌₁𝐸𝑋₁^𝛼𝐼log𝑚<𝑋₁^𝛾()𝐶≤log𝑚+1×loglog𝑚,for𝛼=𝛾,_∞𝑚∑₌₁𝐸𝑋₁^𝛼𝐼log𝑚<𝑋₁^𝛾()×≤log𝑚+1log𝑚^{1−𝛼/𝛾}≤𝑋,for𝛼<𝛾𝐶𝐸₁^𝛼𝑋,for𝛼>𝛾,𝐶𝐸₁^𝛼log𝑋₁𝑋,for𝛼=𝛾,𝐶𝐸₁^𝛾,for𝛼<𝛾<∞, (15) 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 𝐽≤𝐶^∞_𝑛=1𝑛⁻¹𝑏_𝑛^−𝑞𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑌_𝑛𝑖−𝐸𝑌_𝑛𝑖^𝑞≤𝐶^∞_𝑛=1𝑛⁻¹𝑏_𝑛^𝑛−𝑞_𝑖=1𝐸𝑌_𝑛𝑖^𝑞+𝐶^∞_𝑛=1𝑛⁻¹𝑏_𝑛^−𝑞𝑛_𝑖=1𝐸𝑌²_𝑛𝑖^𝑞/2≐𝐽₁+𝐽₂. (16) To prove (14), it suffices to show that 𝐽₁<∞ and 𝐽₂<∞.
For 𝑗≥1 and 𝑛≥2, denote 𝐼_𝑛𝑗=𝑛1≤𝑖≤𝑛∶<𝑎𝑗+1^𝑛𝑖𝛼≤𝑛𝑗. (17) In view of ∑^𝑛_𝑖=1|𝑎_𝑛𝑖|^𝛼≤𝑛, it is easy to see that {𝐼_𝑛𝑗,𝑗≥1} are disjoint and ⋃^∞_𝑗=1𝐼_𝑛𝑗={1≤𝑖≤𝑛∶𝑎_𝑛𝑖≠0}. Hence, we have for all 𝑚≥1 that 𝑛≥^𝑛_𝑖=1𝑎_𝑛𝑖^𝛼=_{{1≤𝑖≤𝑛∶𝑎𝑛𝑖≠0}}𝑎_𝑛𝑖^𝛼=^∞_{𝑗=1𝑖∈𝐼𝑛𝑗}𝑎_𝑛𝑖^𝛼≥𝑛^∞_𝑗=1𝑗+1⁻¹♯𝐼_𝑛𝑗≥𝑛^∞_𝑗=𝑚𝑗+1^−𝑞/𝛼𝑗+1^{𝑞/𝛼−1}♯𝐼_𝑛𝑗≥𝑛^∞_𝑗=𝑚𝑗+1^−𝑞/𝛼(𝑚+1)^{𝑞/𝛼−1}♯𝐼_𝑛𝑗, (18) which implies that for all 𝑚≥1, ^∞_𝑗=𝑚𝑗+1^−𝑞/𝛼♯𝐼_𝑛𝑗≤𝐶𝑚^{1−𝑞/𝛼},𝑛≥2. (19)
By 𝐶_𝑟’s inequality, (13) and (17), we can get that 𝐽₁≤𝐶^∞_𝑛=1𝑛^𝑛−1_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛+𝐶^∞_𝑛=1𝑛⁻¹𝑏_𝑛^𝑛−𝑞_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑖^𝑞𝐼𝑎_𝑛𝑖𝑋_𝑖≤𝑏_𝑛≤𝐶^∞_𝑛=2𝑛⁻¹𝑏_𝑛^𝑛−𝑞_𝑖=1𝐸𝑎_𝑛𝑖𝑋₁^𝑞𝐼𝑎_𝑛𝑖𝑋₁≤𝑏_𝑛≤𝐶^∞_𝑛=2𝑛^{−1−𝑞/𝛼}log𝑛^−𝑞/𝛾×^∞_{𝑗=1𝑖∈𝐼𝑛𝑗}𝐸𝑎_𝑛𝑖𝑋₁^𝑞𝐼𝑋₁≤𝑗+1^1/𝛼log𝑛^1/𝛾≤𝐶^∞_𝑛=2𝑛^{−1−𝑞/𝛼}log𝑛^−𝑞/𝛾×^∞_𝑗=1𝑛^𝑞/𝛼𝑗^−𝑞/𝛼𝐸𝑋₁^𝑞𝐼𝑋₁≤𝑗+1^1/𝛼log𝑛^1/𝛾♯𝐼_𝑛𝑗≤𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝑞/𝛾×^∞_𝑗=1𝑗^−𝑞/𝛼𝐸𝑋₁^𝑞𝐼𝑋₁≤log𝑛^1/𝛾♯𝐼_𝑛𝑗+𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝑞/𝛾×^∞_𝑗=1𝑗^{𝑗−𝑞/𝛼}_𝑘=1𝐸𝑋₁^𝑞𝐼𝑘^1/𝛼log𝑛^1/𝛾<𝑋₁≤(𝑘+1)^1/𝛼log𝑛^1/𝛾♯𝐼_𝑛𝑗≐𝐽₁₁+𝐽₁₂. (20) If 𝛼>𝛾, we have by (19) and 𝐸|𝑋₁|^𝛼<∞ that 𝐽₁₁≤𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝑞/𝛾𝐸𝑋₁^𝑞𝐼𝑋₁≤log𝑛^1/𝛾≤𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝛼/𝛾𝐸𝑋₁^𝛼𝐼𝑋₁≤log𝑛^1/𝛾≤𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝛼/𝛾<∞. (21) If 𝛼≤𝛾, we have by (6) and (19) that 𝐽₁₁≤𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝑞/𝛾𝐸𝑋₁^𝑞𝐼𝑋₁≤log𝑛^1/𝛾≤𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝑞/𝛾×^𝑛_𝑚=2𝐸𝑋₁^𝑞𝐼log(𝑚−1)^1/𝛾<𝑋₁≤log𝑚^1/𝛾=𝐶^∞_𝑚=2𝐸𝑋₁^𝑞𝐼log(𝑚−1)^1/𝛾<𝑋₁≤log𝑚^1/𝛾×^∞_𝑛=𝑚𝑛⁻¹log𝑛^−𝑞/𝛾≤𝐶^∞_𝑚=2log𝑚^{1−𝑞/𝛾}𝐸𝑋₁^𝑞𝐼×log(𝑚−1)^1/𝛾<𝑋₁≤log𝑚^1/𝛾≤𝐶^∞_𝑚=2𝐸𝑋₁^𝛾𝐼log(𝑚−1)^1/𝛾<𝑋₁≤log𝑚^1/𝛾≤𝐶𝐸𝑋₁^𝛾<∞. (22) By (21) and (22), we can get that 𝐽₁₁<∞. Next, we will prove that 𝐽₁₂<∞.
It follows by (6) and (19) again that 𝐽₁₂=𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝑞/𝛾×^∞_𝑘=1𝐸𝑋₁^𝑞𝐼𝑘^1/𝛼log𝑛^1/𝛾<𝑋₁≤(𝑘+1)^1/𝛼log𝑛^1/𝛾×^∞_𝑗=𝑘𝑗^−𝑞/𝛼♯𝐼_𝑛𝑗≤𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝑞/𝛾×^∞_𝑘=1𝑘^{1−𝑞/𝛼}𝐸𝑋₁^𝑞𝐼×𝑘^1/𝛼log𝑛^1/𝛾<𝑋₁≤(𝑘+1)^1/𝛼log𝑛^1/𝛾≤𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝛼/𝛾×^∞_𝑘=1𝐸𝑋₁^𝛼𝐼𝑘^1/𝛼log𝑛^1/𝛾<𝑋₁≤(𝑘+1)^1/𝛼log𝑛^1/𝛾=𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝛼/𝛾𝐸𝑋₁^𝛼𝐼𝑋₁>log𝑛^1/𝛾=𝐶^∞_𝑛=2𝑛⁻¹log𝑛^−𝛼/𝛾×^∞_𝑚=𝑛𝐸𝑋₁^𝛼𝐼log𝑚^1/𝛾<𝑋₁≤log(𝑚+1)^1/𝛾=𝐶^∞_𝑚=2𝐸𝑋₁^𝛼𝐼log𝑚^1/𝛾<𝑋₁≤log(𝑚+1)^1/𝛾×^𝑚_𝑛=2𝑛⁻¹log𝑛^−𝛼/𝛾≤𝐶∑^∞_𝑚=1𝐸𝑋₁^𝛼𝐼log𝑚<𝑋₁^𝛾(),𝐶≤log𝑚+1for𝛼>𝛾,_∞𝑚∑₌₁𝐸𝑋₁^𝛼𝐼log𝑚<𝑋₁^𝛾()𝐶≤log𝑚+1×loglog𝑚,for𝛼=𝛾,_∞𝑚∑₌₁𝐸𝑋₁^𝛼𝐼log𝑚<𝑋₁^𝛾()×≤log𝑚+1log𝑚^{1−𝛼/𝛾}≤𝑋,for𝛼<𝛾𝐶𝐸₁^𝛼𝑋,for𝛼>𝛾,𝐶𝐸₁^𝛼log𝑋₁𝑋,for𝛼=𝛾,𝐶𝐸₁^𝛾,for𝛼<𝛾<∞. (23) By 𝐽₁₁<∞ and 𝐽₁₂<∞, we can get that 𝐽₁<∞.
To prove (14), it suffices to show that 𝐽₂<∞. By 𝐶_𝑟’s inequality, conditions (5) and (6), we can get that 𝐽₂≤𝐶^∞_𝑛=1𝑛^−1𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛^𝑞/2+𝐶^∞_𝑛=1𝑛⁻¹𝑏_𝑛^−𝑞𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑖²𝐼𝑎_𝑛𝑖𝑋_𝑖≤𝑏_𝑛^𝑞/2≤𝐶^∞_𝑛=1𝑛^−1𝑛_𝑖=1𝑏_𝑛^−𝛼𝐸𝑎_𝑛𝑖𝑋₁^𝛼𝐼𝑎_𝑛𝑖𝑋₁≤𝑏_𝑛^𝑞/2≤𝐶𝐸𝑋₁^{𝛼∞𝑞/2}_𝑛=1𝑛⁻¹log𝑛^{−𝛼𝑞/2𝛾}<∞. (24) 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 ∑^𝑛_𝑖=1𝑎_𝑖𝑋_𝑖 of a class of random variables satisfying the Rosenthal-type maximal inequality.

Theorem 3. Let {𝑋_𝑛,𝑛≥1} be a sequence of identically distributed random variables. Let {𝑎_𝑛,𝑛≥1} be a sequence of constants satisfying ∑^𝑛_𝑖=1|𝑎_𝑖|^𝛼=𝑂(𝑛) for some 0<𝛼≤2. 𝐸𝑋_𝑛=0 when 1<𝛼≤2. Let 𝑏_𝑛=𝑛^1/𝛼log^1/𝛾𝑛 for some 𝛾>0. Assume that for any 𝑞≥2, there exists a positive constant 𝐶_𝑞 depending only on 𝑞 such that (4) holds, where 𝑌_𝑛𝑖=−𝑏_𝑛𝐼(𝑎_𝑖𝑋_𝑖<−𝑏_𝑛)+𝑎_𝑖𝑋_𝑖𝐼(|𝑎_𝑖𝑋_𝑖|≤𝑏_𝑛)+𝑏_𝑛𝐼(𝑎_𝑖𝑋_𝑖>𝑏_𝑛) or 𝑌_𝑛𝑖=𝑎_𝑖𝑋_𝑖𝐼(|𝑎_𝑖𝑋_𝑖|≤𝑏_𝑛). Furthermore, suppose that ^∞_𝑛=1𝑛^−1𝑛_𝑖=1𝑃𝑎_𝑖𝑋_𝑖>𝑏_𝑛^𝑞/2<∞ (25) for 𝑌_𝑛𝑖=−𝑏_𝑛𝐼(𝑎_𝑖𝑋_𝑖<−𝑏_𝑛)+𝑎_𝑖𝑋_𝑖𝐼(|𝑎_𝑖𝑋_𝑖|≤𝑏_𝑛)+𝑏_𝑛𝐼(𝑎_𝑖𝑋_𝑖>𝑏_𝑛). If (6) holds, then ^∞_𝑛=11𝑛𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑖𝑋_𝑖>𝜀𝑏_𝑛1<∞,∀𝜀>0,𝑏_𝑛^𝑛_𝑖=1𝑎_𝑖𝑋_𝑖→0a.s.,as𝑛→∞. (26)(27)

Proof. The proof of (26) is the same as that of Theorem 2. So the details are omitted. It suffices to show (27). Denote 𝑆_𝑛=∑^𝑛_𝑖=1𝑎_𝑖𝑋_𝑖 for each 𝑛≥1. It follows by (26) that ∞>^∞_𝑛=1𝑛⁻¹𝑃max_{1≤𝑗≤𝑛}𝑆_𝑗>𝜀𝑏_𝑛=^∞2_𝑖=0^𝑖+1−1_𝑛=2𝑖𝑛⁻¹𝑃max_{1≤𝑗≤𝑛}𝑆_𝑗>𝜀𝑛^1/𝛼log𝑛^1/𝛾≥12^∞_𝑖=1𝑃max_{1≤𝑗≤2𝑖}𝑆_𝑗>𝜀2^{(𝑖+1)/𝛼}log2^𝑖+11/𝛾. (28) By Borel-Cantelli lemma, we obtain that lim_𝑖→∞max_1≤𝑗≤2^𝑖𝑆_𝑗2^{(𝑖+1)/𝛼}log2^𝑖+11/𝛾=0a.s. (29) For all positive integers 𝑛, there exists a positive integer 𝑖₀ such that 2^𝑖0−1≤𝑛<2^𝑖0. We have by (29) that 𝑆_𝑛𝑏_𝑛≤max_{2𝑖0−1≤𝑛<2𝑖0}𝑆_𝑛𝑏_𝑛≤2^2/𝛼max_1≤𝑗≤2^𝑖₀𝑆_𝑗2^{(𝑖0+1)/𝛼}log2^{𝑖0+11/𝛾}𝑖₀+1𝑖₀−1^1/𝛾→0a.s.,as𝑖₀→∞, (30) 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 {𝑋_𝑛,𝑛≥1} be a sequence of identically distributed random variables. Let {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be an array of constants satisfying ∑^𝑛_𝑖=1|𝑎_𝑛𝑖|^𝛼=𝑂(𝑛) for some 0<𝛼≤2. 𝐸𝑋_𝑛=0 when 1<𝛼≤2. Let 𝑏_𝑛=𝑛^1/𝛼log^1/𝛾𝑛 for some 𝛾>0. Assume that for any 𝑞≥2, there exists a positive constant 𝐶_𝑞 depending only on 𝑞 such that 𝐸^𝑛_𝑖=1𝑌_𝑛𝑖−𝐸𝑌_𝑛𝑖^𝑞≤𝐶_𝑞^𝑛_𝑖=1𝐸𝑌_𝑛𝑖^𝑞+^𝑛_𝑖=1𝐸𝑌²_𝑛𝑖^𝑞/2, (31) where 𝑌_𝑛𝑖=−𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖<−𝑏_𝑛)+𝑎_𝑛𝑖𝑋_𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑖|≤𝑏_𝑛)+𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛) or 𝑌_𝑛𝑖=𝑎_𝑛𝑖𝑋_𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑖|≤𝑏_𝑛). Furthermore, suppose that (5) holds for 𝑌_𝑛𝑖=−𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖<−𝑏_𝑛)+𝑎_𝑛𝑖𝑋_𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑖|≤𝑏_𝑛)+𝑏_𝑛𝐼(𝑎_𝑛𝑖𝑋_𝑖>𝑏_𝑛). If (6) satisfies, then ^∞_𝑛=11𝑛𝑃^𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜀𝑏_𝑛<∞,∀𝜀>0. (32)

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]).

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).