Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums of 𝜌-Mixing Random Variables

Abstract

The equivalent conditions of complete convergence are established for weighted sums of 𝜌-mixing random variables with different distributions. Our results extend and improve the Baum and Katz complete convergence theorem. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequence of 𝜌-mixing random variables is obtained.

1. Introduction

Let (Ω,𝐹,𝑃) be a probability space. The random variables we deal with are all defined on (Ω,𝐹,𝑃). Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables. For each nonempty set 𝑆⊂𝑁, write 𝐹_𝑠=𝜎(𝑋_𝑖,𝑖∈𝑆). Given 𝜎-algebras 𝐵, 𝑅 in 𝐹, let 𝜌(𝐵,𝑅)=sup|corr(𝑋,𝑌)|;𝑋∈𝐿₂(𝐵),𝑌∈𝐿₂(𝑅), (1) where corr(𝑋,𝑌)=(𝐸𝑋𝑌−𝐸𝑋𝐸𝑌)/(Var𝑋Var𝑌)^1/2. Define the 𝜌-mixing coefficients by 𝜌(𝑛)=sup𝜌𝐹_𝑆,𝐹_𝑇, (2) where (for a given positive integer 𝑁) this sup is taken over all pairs of nonempty finite subsets 𝑆, 𝑇 of 𝑁 such that dist (𝑆,𝑇)≥𝑛.

Obviously 0≤𝜌(𝑛+1)≤𝜌(𝑛)≤1, 𝑛≥0, and 𝜌(0)=1 except in the trivial case where all of the random variables 𝑋_𝑖 are degenerate.

Definition 1. A sequence of random variables is said to be a 𝜌-mixing sequence of random variables if there exists 𝑘∈𝑁 such that 𝜌(𝑘)<1.
Note that if {𝑋_𝑛,𝑛≥1} is a sequence of independent random variables, then 𝜌(𝑛)=0 for all 𝑛≥1. 𝜌-mixing is similar to 𝜌-mixing, but both are quite different. 𝜌(𝑘) is defined by (2) with index sets restricted to subsets 𝑆 of [1,𝑛] and subsets of 𝑇 of [𝑛+𝑘,∞), 𝑛, 𝑘∈𝑁. On the other hand, 𝜌-mixing sequence assumes the condition 𝜌(𝑘)→0, but 𝜌-mixing sequence assumes the condition that there exists 𝑘∈𝑁 such that 𝜌(𝑘)<1; from this point of view, 𝜌-mixing is weaker than 𝜌-mixing.

The concept of 𝜌-mixing random variables was introduced by Bradley [1] and a number of limit theories for 𝜌-mixing sequences of random variables have been established by many authors. We refer to Bradley [1] for the central limit theorem, Bryc and Smoleński [2], Peligrad and Gut [3], and Utev and Peligrad [4] for moment inequalities, Gan [5], Kuczmaszewska [6], and Wu and Jiang [7] for almost sure convergence, and Cai [8], Zhu [9], An and Yuan [10], Zhou et al. [11], Shen and Hu [12], Guo and Zhu [13], Wang et al. [14], and Sung [15, 16] for complete convergence.

A sequence {𝑋_𝑛,𝑛≥1} of random variables converges completely to the constant 𝐶 if ^∞_𝑛=1𝑃𝑋_𝑛−𝐶>𝜖<∞forany𝜖>0. (3)

In view of the Borel-Cantelli lemma, this implies that 𝑋_𝑛→𝐶 almost surely. Hence, complete convergence is one of the most important problems in probability theory. Since the concept of complete convergence was introduced by Hsu and Robbins [17], there have been many authors who devoted the study to complete convergence for independent and identically distributed random variables. One of the most important results is Baum and Katz theorem [18]. The theorem was further generalized and extended in different ways. Katz [19] and Chow [20] formed the following generalization with a normalization of Marcinkiewicz-Zygmund type theorem for the strong law of large numbers.

Theorem 2 (see [21]). Let 𝛼𝑝≥1, 𝛼>1/2, and let {𝑋_𝑛,𝑛≥1} be a sequence of independent and identically distributed random variables. If 𝑝≥1, assume that 𝐸𝑋₁=0. Then the following statements are equivalent:(i)𝐸|𝑋₁|^𝑝<∞; (ii)∑^∞_𝑛=1𝑛^𝛼𝑝−2𝑃(max_{1≤𝑗≤𝑛}|∑^𝑗_𝑖=1𝑋_𝑖|>𝜖𝑛^𝛼)<∞ for all 𝜖>0.

In many stochastic models, the assumption of independence among random variables is not plausible. So it is necessary to extend the concept of independence to dependence cases. Peligrad and Gut [3] extended this result from independent and identically distributed case to the case of 𝜌-mixing random variables with identical distribution. But they did not prove whether the result of Theorem 2 of the case 𝛼𝑝=1 holds for 𝜌-mixing sequence. In practical applications it is difficult to check the independence of a sample or the samples are not independent observations. Therefore, in recent investigations limit theorems are very often considered for sequences of dependent random variables. Recently, a number of limit theorems for dependent random variables have been established by many authors. We can refer to Sung [22], Wu and Jiang [23], Wu [24], and Shen [25].

Let {𝑋_𝑛,𝑛≥1} be a sequence of identically distributed random variables and let {𝑎_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} be an array of constants. The strong convergence results for weighted sums ∑^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘 have been studied by many authors; see, for example, Cuzick [26], Choi and Sung [27], Bai and Cheng [28], Chen and Gan [29], and so forth. Many useful linear statistics are weighted sums. Examples include least squares estimators, nonparametric regression function estimators, and jackknife estimates.

Inspired by Theorem 2.1 of Kuczmaszewska [30], our main purpose in this work is to extend the complete convergence for weighted sums ∑^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘 of independent and identically distributed random variables to the case of 𝜌-mixing random variables. However, our proven methods are different from the ones by Kuczmaszewska [30]; by applying inequality (13) of Lemma 10 our proof is much simpler than the one by Kuczmaszewska. Our proof of necessary condition (using Lemma 10) is original. We provide sufficient and necessary conditions of complete convergence for weighted sums of 𝜌-mixing random variables with different distributions. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of 𝜌-mixing random variables are obtained. In addition, our main results extend and improve the corresponding results of Peligrad and Gut [3].

Throughout this paper, the symbol 𝐶 denotes a positive constant which is not necessarily the same one in each appearance, 𝑎_𝑛=𝑂(𝑏_𝑛) will mean 𝑎_𝑛≤𝐶(𝑏_𝑛) for sufficiently large 𝑛, 𝑎_𝑛≪𝑏_𝑛 will mean 𝑎_𝑛=𝑂(𝑏_𝑛), and 𝐼(𝐴) is the indicator function of event 𝐴.

2. Main Results

Now we state our main results of this paper. The proofs will be given in Section 3.

Theorem 3. Let 𝑋 be a random variable and let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜌-mixing random variables satisfying the condition 1𝑛^𝑛_𝑘=1𝑃𝑋_𝑘>𝑥=𝐶𝑃(|𝑋|>𝑥) (4) for all 𝑥>0, all 𝑛≥1, and some positive constant 𝐶. Let {𝑎_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} be a sequence of real numbers such that 𝑎_𝑛𝑘≍𝑛^−𝛼,∀1≤𝑘≤𝑛,𝑛≥1, (5) where 𝑎≍𝑏 means 𝑎=𝑂(𝑏) and 𝑏=𝑂(𝑎). Let 𝛼𝑝≥1, 𝛼>1/2, and if 𝛼≤1, assume that 𝐸𝑋_𝑛=0, 𝑛≥1. Then the following statements are equivalent: (i)𝐸|𝑋|^𝑝<∞, (ii)∑^∞_𝑛=1𝑛^𝛼𝑝−2𝑃(max_{1≤𝑘≤𝑛}|∑^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖|>𝜖)<∞𝑓𝑜𝑟𝑎𝑙𝑙𝜖>0.

Remark 4. When proving the limit theorem of 𝜌-mixing random variables with different distributions, many authors apply the condition of {𝑋_𝑛,𝑛≥1} being stochastically dominated by 𝑋, that is, for some constant 𝐶>0, 𝑃(|𝑋_𝑛|≥𝑥)≤𝐶𝑃(|𝑋|≥𝑥), for all 𝑥≥0, 𝑛≥1, which implies that (1/𝑛)∑^𝑛_𝑘=1𝑃(|𝑋_𝑘|>𝑥)≤𝐶𝑃(|𝑋|>𝑥), but the converse is not true. Hence our condition of (4) is weaker than the condition of stochastic dominance.

When {𝑋_𝑛,𝑛≥1} is a sequence of 𝜌-mixing identically distributed random variables and 𝑎_𝑛𝑖=𝑛^−𝛼, for all 1≤𝑖≤𝑛, 𝑛≥1, then Theorem 3 becomes Baum and Katz complete convergence theorem as follows.

Corollary 5. Let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜌-mixing identically distributed random variables. Let 𝛼𝑝≥1, 𝛼>1/2, and if 𝛼≤1, assume that 𝐸𝑋_𝑛=0, 𝑛≥1. Then the following statements are equivalent:(i)𝐸|𝑋₁|^𝑝<∞; (ii)∑^∞_𝑛=1𝑛^𝛼𝑝−2𝑃(max_{1≤𝑗≤𝑛}|∑^𝑗_𝑖=1𝑋_𝑖|>𝜖𝑛^𝛼)<∞ for all 𝜖>0.

Remark 6. Corollary 5 not only generalizes Theorem 2 to 𝜌-mixing case, but also extends Theorem 2 of Peligrad and Gut [3] to the case 𝛼𝑝=1. Therefore, Corollary 5 improves and extends the well-known Baum and Katz theorem.

An and Yuan [10, Theorem 2] presented a Marcinkiewicz-Zygmund type strong law of large numbers for 𝜌-mixing sequence. We find that the proof of their Theorem 2 is wrong because the theorem is based on Theorem 1 [10]. However, the author thinks that their proofs of Theorem 1 have a little problem, since condition (1.2) does not hold for the array with {𝑎_𝑛𝑖,1≤𝑖≤𝑛}. An and Yuan [10, Theorem 1] proved the implication (i) ⇒ (ii) under condition (1.3) and proved the converse under conditions (1.2) and (1.3). However, the array satisfying both (1.2) and (1.3) does not exist. Noting that ♯𝐴_𝑛𝑘/(𝑘+1)≤∑^𝑛_𝑖=1|𝑎_𝑛𝑖|^𝑝≤𝑂(𝑛^𝛿), we have that 𝑛𝑒^−1/𝑘≤♯𝐴_𝑛𝑘≤(𝑘+1)𝑂(𝑛^𝛿). But this does not hold when 𝑘 is fixed and 𝑛 is large enough. In this paper, we obtain a new complete convergence result for weighted sums of 𝜌-mixing random variables without assumption of identical distribution. Our result generalizes and sharpens the result of An and Yuan [10]. The following corollary provides the Marcinkiewicz-Zygmund type strong law of large numbers of 𝜌-mixing random variables without assumption of identical distribution.

Corollary 7. Let 𝑋 be a random variable and let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜌-mixing random variables satisfying the condition 1𝑛^𝑛_𝑘=1𝑃𝑋_𝑘>𝑥=𝐶𝑃(|𝑋|>𝑥) (6) for all 𝑥>0, all 𝑛≥1, and some positive constant 𝐶.  𝐸|𝑋|^𝑝<∞ for some 0<𝑝<2 and if 1≤𝑝<2, assume that 𝐸𝑋_𝑛=0, 𝑛≥1. Then, for any 𝜖>0, ^∞_𝑛=1𝑛⁻¹𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖>𝜖𝑛^1/𝑝1<∞,𝑛^1/𝑝𝑛_𝑖=1𝑋_𝑖⟶0𝑎.𝑠.,𝑛⟶∞. (7)

3. Proof of Main Results

The following lemmas are useful for the proof of the main results.

Lemma 8 (see [4]). Suppose 𝐾 is a positive integer, 0≤𝑟<1, and 𝑞≥2. Then there exists a constant 𝐷=𝐷(𝐾,𝑟,𝑞) such that the following statement holds.
If {𝑋_𝑖,𝑖≥1} is a sequence of random variables such that 𝜌(𝐾)≤𝑟 and 𝐸𝑋_𝑛=0 and 𝐸|𝑋_𝑖|^𝑞<∞ for all 𝑖≥1, then, for every 𝑛≥1, 𝐸max_{1≤𝑖≤𝑛}^𝑖_𝑗=1𝑋_𝑗^𝑞≤𝐷^𝑛_𝑖=1𝐸𝑋_𝑖^𝑞+^𝑛_𝑖=1𝐸𝑋²_𝑖^𝑞/2. (8)

Lemma 9 (see [30]). Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables which is weakly mean dominated by a random variable 𝑋; that is, for all 𝑥≥0 and some positive constant 𝐶>0, 1𝑛^𝑛_𝑘=1𝑃𝑋_𝑘>𝑥≤𝐶𝑃(|𝑋|>𝑥). (9)
Then for any 𝑢>0, 𝑡>0, and 𝑛≥1, the following three statements hold: 𝐼𝑓𝐸|𝑋|^𝑢1<∞,𝑡ℎ𝑒𝑛𝑛^𝑛_𝑘=1𝐸𝑋_𝑘^𝑢≤𝐶𝐸|𝑋|^𝑢,1𝑛^𝑛_𝑘=1𝐸𝑋_𝑘^𝑢𝐼𝑋_𝑘𝐸|𝑋|≤𝑡≤𝐶^𝑢𝐼(|𝑋|≤𝑡)+𝑡^𝑢𝑃(|𝑋|>𝑡),1𝑛^𝑛_𝑘=1𝐸𝑋_𝑘^𝑢𝐼𝑋_𝑘>𝑡≤𝐶𝐸|𝑋|^𝑢𝐼(|𝑋|>𝑡). (10)(11)(12)

Lemma 10. Let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜌-mixing random variables. Then there exists a positive constant 𝐶 such that, for any 𝑥≥0 and all 𝑛≥1, 12−𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘>𝑥^𝑛_𝑘=1𝑃𝑋_𝑘≤𝐶>𝑥2+1𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘>𝑥. (13)

Proof of Lemma 10. Since {max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝑥}=⋃^𝑛_𝑘=1{|𝑋_𝑘|>𝑥,max_{1≤𝑗≤𝑘−1}|𝑋_𝑗|≤𝑥}, we have ^𝑛_𝑘=1𝑃𝑋_𝑘=>𝑥_𝑛𝑘=1𝑃𝑋_𝑘>𝑥,max_{1≤𝑗≤𝑘−1}𝑋_𝑗+≤𝑥_𝑛𝑘=1𝑃𝑋_𝑘>𝑥,max_{1≤𝑗≤𝑘−1}𝑋_𝑗>𝑥=𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘+>𝑥_𝑛𝑘=1𝑃𝑋_𝑗>𝑥,max_{1≤𝑗≤𝑘−1}𝑋_𝑗>𝑥. (14)
Note that ^𝑛_𝑘=1𝑃𝑋_𝑘>𝑥,max_{1≤𝑗≤𝑘−1}𝑋_𝑗=>𝑥_𝑛𝑘=1𝐸𝐼𝑋_𝑘>𝑥𝐼max_{1≤𝑗≤𝑘−1}𝑋_𝑗>𝑥≤𝐸^𝑛_𝑘=1𝐼𝑋_𝑘>𝑥−𝐸𝐼𝑋_𝑘>𝑥×𝐼max_{1≤𝑗≤𝑛}𝑋_𝑗+>𝑥_𝑛𝑘=1𝑃𝑋_𝑘>𝑥𝑃max_{1≤𝑗≤𝑛}𝑋_𝑗>𝑥≜𝐽₁+𝐽₂. (15)
Obviously, by Lemma 8, we get 𝐸^𝑛_𝑘=1𝑋_𝑘^𝑞≤𝐶^𝑛_𝑘=1𝐸𝑋_𝑘^𝑞+^𝑛_𝑘=1𝐸𝑋²_𝑘^𝑞/2. (16)
Combining with the Cauchy-Schwarz inequality and (16), we obtain 𝐽₁=𝐸^𝑛_𝑘=1𝐼𝑋_𝑘>𝑥−𝐸𝐼𝑋_𝑘>𝑥×𝐼max_{1≤𝑗≤𝑛}𝑋_𝑗≤𝐸>𝑥_𝑛𝑘=1𝐼𝑋_𝑘>𝑥−𝐸𝐼𝑋_𝑘>𝑥²×𝑃max_{1≤𝑗≤𝑛}𝑋_𝑗>𝑥^1/2≤𝐶^𝑛_𝑘=1𝑃𝑋_𝑘>𝑥𝑃max_{1≤𝑗≤𝑛}𝑋_𝑗>𝑥^1/2≤12^𝑛_𝑘=1𝑃𝑋_𝑘+𝐶>𝑥2𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘>𝑥. (17)
Now, we substitute (17) into (15) and then into (14), which implies that (13) holds.

Consequently, we prove our main results.

Proof of Theorem 3. First, we prove that (i)⇒(ii).
Note that 𝑎_𝑛𝑖=𝑎⁺_𝑛𝑖−𝑎⁻_𝑛𝑖, where 𝑎⁺_𝑛𝑖=max{0,𝑎_𝑛𝑖} and 𝑎⁻_𝑛𝑖=max{0,−𝑎_𝑛𝑖}. To prove (ii) it suffices to show that ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎^±_𝑛𝑖𝑋_𝑖>𝜖<∞,∀𝜖>0. (18)
Thus, without loss of generality, we may assume that 𝑎_𝑛𝑖>0 for all 1≤𝑖≤𝑛, 𝑛≥1.
For fixed 𝑛≥1, denote that 𝑋_𝑛𝑖=𝑋_𝑖𝐼𝑋_𝑖≤𝑛^𝛼,1≤𝑖≤𝑛. (19)
Firstly, we show that max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖⟶0,𝑛⟶∞. (20)
If 1/2<𝛼≤1, by 𝐸𝑋_𝑛=0, (i), (5), (12) of Lemma 9, and 𝛼𝑝≥1, we have max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖≤^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖≪𝑛^𝑛−𝛼_𝑖=1𝐸𝑋_𝑛𝑖=𝑛^𝑛−𝛼_𝑖=1𝐸𝑋_𝑖𝐼𝑋_𝑖>𝑛^𝛼≤𝑛^1−𝛼𝐸|𝑋||𝑋|𝑛^𝛼𝑝−1𝐼|𝑋|>𝑛^𝛼≪𝑛^1−𝛼𝑝𝐸|𝑋|^𝑝𝐼|𝑋|>𝑛^𝛼⟶0,𝑛⟶∞. (21)
If 𝛼>1, 𝑝≥1,   by (5), (11) of Lemma 9, Markov inequality, and 𝐸|𝑋|<∞ from (i), we get max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖≪𝑛^𝑛−𝛼_𝑖=1𝐸𝑋_𝑖𝐼𝑋_𝑖≤𝑛^𝛼≪𝑛^1−𝛼𝐸|𝑋|𝐼|𝑋|≤𝑛^𝛼+𝑛^𝛼𝑃|𝑋|>𝑛^𝛼≪𝑛^1−𝛼𝐸|𝑋|⟶0,𝑛⟶∞. (22)
If 𝛼>1, 0<𝑝<1, by 𝛼𝑝≥1, we can get lim_𝑛→∞𝑛𝑃(|𝑋|>𝑛^𝛼)=0, and thus max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖≪𝑛^𝑛−𝛼_𝑖=1𝐸𝑋_𝑖𝐼𝑋_𝑖≤𝑛^𝛼≪𝑛^1−𝛼𝐸|𝑋|𝐼|𝑋|≤𝑛^𝛼+𝑛^𝛼𝑃|𝑋|>𝑛^𝛼≪𝑛^𝑛1−𝛼_𝑖=1𝐸|𝑋|𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼+𝑛𝑃|𝑋|>𝑛^𝛼. (23)
Note that, if 𝛼𝑝≥1, we have ^∞_𝑖=1𝑖^1−𝛼𝐸|𝑋|𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼≤^∞_𝑖=1𝑖^1−𝛼𝑝𝐸|𝑋|^𝑝𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼≤^∞_𝑖=1𝐸|𝑋|^𝑝𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼=𝐸|𝑋|^𝑝<∞. (24)
Hence, by Kronecker lemma and (23), we obtain max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖≪𝑛^𝑛1−𝛼_𝑖=1𝐸|𝑋|𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼⟶0,𝑛⟶∞. (25)
From (21), (22), and (25) we can get (20) immediately. Hence, for all 𝑛 sufficiently large and any 𝜖>0, we have max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖<𝜖2. (26)
It is easy to check that for all 𝑛 sufficiently large max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖⊂>𝜖_𝑛𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜖∪max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖⊂>𝜖_𝑛𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜖∪max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜖2≜𝐴_𝑛∪𝐵_𝑛, (27) which implies that for all 𝑛 sufficiently large 𝑃max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜖≤𝑃𝐴_𝑛+𝑃𝐵_𝑛. (28)
Therefore, in order to prove (ii), we only need to prove that ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃𝐴_𝑛<∞,_∞𝑛=1𝑛^𝛼𝑝−2𝑃𝐵_𝑛<∞. (29)(30)
By (4), (5), and 𝛼𝑝≥1, we can get that ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃𝐴_𝑛≤^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2}_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑖≤>𝜖_∞𝑛=1𝑛^{𝑛𝛼𝑝−2}_𝑖=1𝑃𝑋_𝑖>𝜖𝑎⁻¹_𝑛𝑖≥𝐶𝜖𝑛^𝛼≪^∞_𝑛=1𝑛^𝛼𝑝−1𝑃|𝑋|>𝐶𝜖𝑛^𝛼=^∞_𝑛=1𝑛^{∞𝛼𝑝−1}_𝑖=𝑛𝑃𝐶𝜖𝑖^𝛼<|𝑋|≤𝐶𝜖(𝑖+1)^𝛼=^∞𝑖_𝑖=1𝑛=1𝑛^𝛼𝑝−1𝑃𝐶𝜖𝑖^𝛼<|𝑋|≤𝐶𝜖(𝑖+1)^𝛼≤^∞_𝑖=1𝑖^𝛼𝑝𝑃𝐶𝜖𝑖^𝛼<|𝑋|≤𝐶𝜖(𝑖+1)^𝛼≪𝐸|𝑋|^𝑝<∞. (31)
That is, (29) holds. Thus, it remains to prove (30).
Since {𝑋_𝑛,𝑛≥1} is a sequence of 𝜌-mixing random variables and 𝑋_𝑛𝑖=𝑋_𝑖𝐼(|𝑋_𝑖|≤𝑛^𝛼), 1≤𝑖≤𝑛, thus {𝑋_𝑛𝑖−𝐸𝑋_𝑛𝑖,1≤𝑖≤𝑛} is still a sequence of 𝜌-mixing random variables with 𝐸(𝑋_𝑛𝑖−𝐸𝑋_𝑛𝑖)=0. By Markov inequality and Lemma 8, let 𝑞≥2. Then, ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃𝐵_𝑛=^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜖2≪^∞_𝑛=1𝑛^𝛼𝑝−2𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖^𝑞≪^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼𝑞}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑛𝑖−𝐸𝑋_𝑛𝑖^𝑞≪^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼𝑞𝑛}_𝑖=1𝐸𝑋²_𝑛𝑖^𝑞/2+^𝑛_𝑖=1𝐸𝑋_𝑛𝑖^𝑞≜𝐼₁+𝐼₂. (32)
When 𝑝≥2, taking 𝑞>max{(𝛼𝑝−1)/(𝛼−1/2),2}, then 𝛼𝑝−2−𝛼𝑞+𝑞/2<−1, and by Jensen inequality and (11) of Lemma 9, we have 𝐼₁=^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼𝑞𝑛}_𝑖=1𝐸𝑋²_𝑛𝑖^𝑞/2≪^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼𝑞+𝑞/2}𝐸𝑋²𝐼|𝑋|≤𝑛^𝛼+𝑛^2𝛼𝑃|𝑋|>𝑛^𝛼𝑞/2≪^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼𝑞+𝑞/2}𝐸𝑋^2𝑞/2≪^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼𝑞+𝑞/2}<∞. (33)
Taking 𝑞>𝑝, we have by (11) of Lemma 9 that 𝐼₂=^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2−𝛼𝑞}_𝑖=1𝐸𝑋_𝑛𝑖^𝑞≪^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼𝑞+1}𝐸|𝑋|^𝑞𝐼|𝑋|≤𝑛^𝛼+𝑛^𝛼𝑞𝑃|𝑋|>𝑛^𝛼≪^∞_𝑛=1𝑛^{𝛼𝑝−1−𝛼𝑞}𝐸|𝑋|^𝑞𝐼|𝑋|≤𝑛^𝛼+^∞_𝑛=1𝑛^𝛼𝑝−1𝑃|𝑋|>𝑛^𝛼≪^∞_𝑛=1𝑛^{𝑛𝛼𝑝−1−𝛼𝑞}_𝑖=1𝐸|𝑋|^𝑞𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼+𝐸|𝑋|^𝑝≪^∞∞_{𝑖=1𝑛=𝑖}𝑛^{𝛼𝑝−1−𝛼𝑞}𝐸|𝑋|^𝑞𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼≪^∞_𝑖=1𝑖^{𝛼𝑝−𝛼𝑞}𝐸|𝑋|^𝑞𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼=^∞_𝑖=1𝑖^{𝛼𝑝−𝛼𝑞}𝐸|𝑋|^𝑝|𝑋|^𝑞−𝑝𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼≤^∞_𝑖=1𝐸|𝑋|^𝑝𝐼(𝑖−1)^𝛼<|𝑋|≤𝑖^𝛼=𝐸|𝑋|^𝑝<∞. (34)
When 𝑝<2, then, taking 𝑞=2, by (32), we get ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃𝐵_𝑛≪^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2−2𝛼}_𝑖=1𝐸𝑋_𝑛𝑖². (35)
Similarly to the proof of inequality (34), we obtain ^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2−2𝛼}_𝑖=1𝐸𝑋_𝑛𝑖²<∞, (36) which implies that ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃𝐵_𝑛<∞. (37)
Now, we prove the converse. To prove that (ii) implies (i), it suffices to show that ^∞_𝑛=1𝑛^𝛼𝑝−1𝑃|𝑋|>𝜖𝑛^𝛼<∞,∀𝜖>0. (38)
Noting that max_{1≤𝑖≤𝑛}𝑎_𝑛𝑖𝑋_𝑖≤max_{1≤𝑖≤𝑛}^𝑖_𝑗=1𝑎_𝑛𝑗𝑋_𝑗+max_{1≤𝑖≤𝑛}^𝑖−1_𝑗=1𝑎_𝑛𝑗𝑋_𝑗, (39) then from (ii) and (5), we have ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑖≤𝑛}𝑋_𝑖>𝜖𝑛^𝛼≪^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑖≤𝑛}𝑎_𝑛𝑖𝑋_𝑖≪>𝜖_∞𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑖≤𝑛}^𝑖_𝑗=1𝑎_𝑛𝑗𝑋_𝑗>𝜖<∞. (40)
Combining with the condition of 𝛼𝑝≥1, 𝑃max_{1≤𝑖≤𝑛}𝑋_𝑖>𝜖𝑛^𝛼⟶0,𝑛⟶∞. (41)
Thus, for sufficiently large 𝑛, 𝑃max_{1≤𝑖≤𝑛}𝑋_𝑖>𝜖𝑛^𝛼<12. (42)
Therefore, by applying Lemma 10, it is easy to see that ^𝑛_𝑖=1𝑃𝑋_𝑖>𝜖𝑛^𝛼≪𝑃max_{1≤𝑖≤𝑛}𝑋_𝑖>𝜖𝑛^𝛼, (43) which, together with the conditions of (4) and (40), gives ^∞_𝑛=1𝑛^𝛼𝑝−1𝑃|𝑋|>𝜖𝑛^𝛼≪^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2}_𝑖=1𝑃𝑋_𝑖>𝜖𝑛^𝛼<∞, (44) which implies that (i) holds. This completes the proof of Theorem 3.

Proof of Corollary 7. Taking 𝛼=1/𝑝 and 𝑎_𝑛𝑖=𝑛^−𝛼, for all 1≤𝑖≤𝑛,𝑛≥1, in Theorem 3, we can get (7) immediately; thus ∞>^∞_𝑛=1𝑛⁻¹𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖>𝜖𝑛^1/𝑝=^∞2_𝑖=0^𝑖+1−1_𝑛=2𝑖𝑛⁻¹𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖>𝜖𝑛^1/𝑝≥12^∞_𝑖=1𝑃max_{1≤𝑗≤2𝑖}^𝑗_𝑖=1𝑋_𝑖>𝜖2^{(𝑖+1)/𝑝}. (45)
It follows from Borel-Cantelli lemma that 𝑃max_{1≤𝑗≤2𝑖}^𝑗_𝑖=1𝑋_𝑖>𝜖2^{(𝑖+1)/𝑝},i.o.=0. (46)
Hence, lim_𝑖→∞12^{(𝑖+1)/𝑝}max_{1≤𝑗≤2𝑖}^𝑗_𝑖=1𝑋_𝑖=0a.s. (47)
For all positive integers 𝑛, there exists a nonnegative integer 𝑖₀ such that 2^𝑖0−1≤𝑛<2^𝑖0. We have by (47) that max_{2𝑖0−1≤𝑛≤2𝑖0}1𝑛^1/𝑝𝑛_𝑖=1𝑋_𝑖≤12^{(𝑖0−1)/𝑝}max_{1≤𝑗≤2𝑖0}^𝑗_𝑖=1𝑋_𝑖≤2^2/𝑝2^{(𝑖0+1)/𝑝}max_{1≤𝑗≤2𝑖0}^𝑗_𝑖=1𝑋_𝑖⟶0a.s.,𝑖₀⟶∞, (48) which implies that 1𝑛^1/𝑝𝑛_𝑖=1𝑋_𝑖⟶0a.s.,𝑛⟶∞. (49)
The proof of Corollary 7 is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the editor and the anonymous referees for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. The research is supported by the National Natural Science Foundation of China (11226200) and the Natural Science Research Project of Anhui Province (KJ2013Z265).