On Complete Convergence and Strong Law for Weighted Sums of i.i.d. Random Variables

Abstract

We improve and generalize the result of Stout (1974, Theorem 4.1.3). In particular, the sharp moment conditions are obtained and some well-known results can be obtained as special cases of the main result. The method of the proof is completely different from that in Stout. We also improve and generalize Li et al. (1995) strong law for weighted sums of i.i.d. random variables.

1. Introduction and the Main Result

Stout [1] obtained the following celebrated result.

Theorem A (see Theorem  4.1.3 in [1] or p. 1556 in Stout [2]). Let {𝑋,𝑋_𝑛≥1} be a sequence of independent and identically distributed random variables with 𝐸𝑋=0 and 𝐸|𝑋|^2/𝛼<∞ for some 0<𝛼≤1. Suppose that {𝑎_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} is a sequence of constants with max_{1≤𝑘≤𝑛}𝑎_𝑛𝑘≤𝐾𝑛^−𝛼 (1) for some constant 0<𝐾<∞ and lim_𝑛→∞log𝑛^𝑛_𝑘=1𝑎²_𝑛𝑘=0. (2) Then ^∞_𝑛=1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘>𝜀<∞,∀𝜀>0. (3)

Formula (3) is called complete convergence and this concept was introduced by Hsu and Robbins [3]. Sung [4] and Cheng and Wang [5] extended Theorem A to random elements taking values in a Banach space. Wu [6] and Sung [7] extended Theorem A to negatively dependent random variables. It should be pointed out that they all used some exponential inequalities to prove their result and hence the proofs are similar to that of Theorem A except for more computational complexity.

When 1/2<𝛼≤1, (1) clearly implies (2). Set 𝑎_𝑛𝑘=𝑛^−𝛼 for 1≤𝑘≤𝑛 and 𝑛≥1. Then (3) reduces to ^∞_𝑛=1𝑃^𝑛_𝑘=1𝑋_𝑘>𝜀𝑛^𝛼<∞,∀𝜀>0, (4) which is equivalent to 𝐸|𝑋|^2/𝛼<∞ and 𝐸𝑋=0 by Katz [8] and Baum and Katz [9]. Hence, the moment conditions of Theorem A are sharp when 1/2<𝛼≤1.

Next, we consider the case of 0<𝛼≤1/2. Lai [10] showed that ^∞_𝑛=1𝑃^𝑛_𝑘=1𝑋_𝑘>𝜀𝑛log𝑛<∞ (5) for 𝜀>√2 provided 𝐸𝑋=0, 𝐸𝑋²=1, and 𝐸𝑋⁴/log²|𝑋|<∞. Hence, for any 𝜀>0, ^∞_𝑛=1𝑃^𝑛_𝑘=1𝑋_𝑘>𝜀𝑐_𝑛𝑛log𝑛<∞ (6) if 𝑐_𝑛→∞. Set 𝑎_𝑛𝑘=(𝑐_𝑛𝑛log𝑛)⁻¹ for 1≤𝑘≤𝑛 and 𝑛≥1. Then (1) and (2) hold for 𝛼=1/2. Note that the moment conditions are weaker than those of Theorem A when 𝛼=1/2. Hence, the moment conditions of Theorem A may not be optimal for a special case of weighted sums. But it is not known whether the moment conditions of Theorem A are not optimal when 0<𝛼≤1/2.

In this paper, we will discuss the optimized moment conditions of Theorem A when 0<𝛼≤1/2. We obtain a more generalized complete convergence result for weighted sums which includes the result of Lai [10]. Our method used is completely different from those in Lai [10] and Stout [1].

Li et al. [11] obtained the following celebrated result.

Theorem B (see Theorem  3.1 in [11]). Let {𝑋,𝑋_𝑛≥1} be a sequence of independent and identically distributed random variables with 𝐸𝑋=0 and 𝐸|𝑋|^1/𝛼<∞ for some 0<𝛼≤1. Suppose that {𝑎_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} is a sequence of constants with (1) for some constant 0<𝐾<∞ and lim_𝑛→∞𝑛^𝛾𝑛_𝑘=1𝑎²_𝑛𝑘=0, (7) where 𝛾>2𝛼−1 if 1/2≤𝛼≤1 and 𝛾>0 if 0<𝛼<1/2. Then lim^𝑛_{𝑛→∞𝑘=1}𝑎_𝑛𝑘𝑋_𝑘=0𝑎.𝑠. (8)

Theorem B has been extended and improved by many authors. Jing and Liang [12] extended and improved to negatively associated (NA) random variables, Budsaba et al. [13] to certain types of 𝑈-statistics bases on this kind of weighted sums of NA random variables, and Thanh and Yin [14] to the random weighted sums. In particular, under the condition limsup_𝑛→∞log𝑛^𝑛_𝑘=1𝑎²_𝑛𝑘<∞, (9) Jing and Liang [12] showed that limsup_𝑛→∞^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘<∞𝑎.𝑠. (10) when 0<𝛼<1/2.

Is it possible to find the sharp bound of (10)? In this paper, we will give a definite answer to the question under more general case.

Throughout this paper, 𝑐 represents a positive constant which may vary in different places, [𝑥] denotes the integer part of 𝑥, and 𝑓(𝑥)∼𝑔(𝑥) means 𝑓(𝑥)/𝑔(𝑥)→1 as 𝑥→∞. It proves convenient to define log𝑥=max{1,ln𝑥}, where ln𝑥 denotes the natural logarithm.

Now, we are ready to state the main results, and the proofs will be given in the next section.

Theorem 1. Let 𝑟>0,0<𝛼≤1/2, 𝑔(𝑥) be an increasing and regular varying function at infinity with index 𝛼 and let ℎ(𝑥) be the inverse function of 𝑔(𝑥). Let {𝑋,𝑋_𝑛≥1} be a sequence of independent and identically distributed random variables with 𝐸𝑋=0, 𝐸𝑋²=1, and 𝐸ℎ^𝑟(|𝑋|)|𝑋|²/logℎ(|𝑋|)<∞. Suppose that {𝑎_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} is a sequence of constants such that max_{1≤𝑘≤𝑛}𝑎_𝑛𝑘≤𝐾𝑔⁻¹(𝑛) (11) for some constant 𝐾>0 and lim_𝑛→∞log𝑛^𝑛_𝑘=1𝑎²_𝑛𝑘=𝜌 (12) for some constant 0≤𝜌<∞. Then ^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘>𝜀<∞if𝜀>2𝑟𝜌,=∞if0<𝜀<2𝑟𝜌. (13)

Remark 2. Recall that a measurable function 𝑓(𝑥) is said to be regularly varying at infinity with index 𝛽 if it is positive on [0,∞) and lim_𝑥→∞𝑓(𝜆𝑥)𝑓(𝑥)=𝜆^𝛽,∀𝜆>0. (14) We refer to Bingham et al. [15] for other equivalent definitions and for detailed and comprehensive study of properties of regularly varying functions. For example, if 𝛽>−1, ^𝑛_𝑘=1𝑓(𝑘)≤𝑐_𝛽𝑛𝑓(𝑛), (15) and if 𝛽<−1, ^∞_𝑘=𝑛𝑓(𝑘)≤𝐶_𝛽𝑛𝑓(𝑛), (16) where 𝑐_𝛽>0 and 𝐶_𝛽>0 are constants depending only on 𝛽.

Remark 3. When 𝑟=1 and 𝑔(𝑥)=𝑥^𝛼, the last moment condition of Theorem 1 is reduced to 𝐸|𝑋|^1/𝛼+2/log|𝑋|<∞. When 0<𝛼≤1/2, 1/𝛼+2≤2/𝛼 and so the moment conditions in Theorem 1 are strictly weaker than those in Theorem A.

Remark 4. Let 𝑎_𝑛𝑘=𝑔⁻¹(𝑛) if 1≤𝑘≤[𝑔²(𝑛)/log𝑛] and 𝑎_𝑛𝑘=0 if 𝑘>[𝑔²(𝑛)/log𝑛]. Then, it is easy to show that lim_𝑛→∞log𝑛^𝑛_𝑘=1𝑎_𝑛𝑘²=1. (17) Hence, (11) and (12) hold. Under the moment conditions of Theorem 1, ^∞_𝑛=1𝑛^𝑟−1𝑃^{[𝑔2(𝑛)/log𝑛]}_𝑘=1𝑋_𝑘>𝜀𝑔(𝑛)<∞ (18) for all 𝜀>√2𝑟. On the other hand, it is easy to show that if the above formula holds for some 𝜀>0, then 𝐸ℎ^𝑟(|𝑋|)|𝑋|²/logℎ(|𝑋|)<∞ by the similar argument as in Lai [10]. Thus, the moment conditions of Theorem 1 are sharp in the sense that the moment conditions on 𝑋 cannot be weakened.

Remark 5. Let 𝑔(𝑥)=𝑥log𝑥 and 𝑎_𝑛𝑘=𝑔⁻¹(𝑛) for 1≤𝑘≤𝑛 and 𝑛≥1. Then, by Theorem 1, the moment conditions 𝐸𝑋=0, 𝐸𝑋²=1, and 𝐸(𝑋²/log|𝑋|)^𝑟+1<∞ imply that ^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑋_𝑘>𝜀√𝑛log𝑛<∞if𝜀>√2𝑟,=∞if0<𝜀<2𝑟. (19) So, the sufficient part of Theorem 3 in Lai [10] is a special case of Theorem 1.
By Theorem 1 and Borel-Cantelli lemma, we have the following corollary.

Corollary 6. Under the conditions of Theorem 1, let 𝑟=1, {𝑋_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} be an array of independent random variables with the same distribution as 𝑋. Then limsup_𝑛→∞^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑛𝑘=2𝜌𝑎.𝑠. (20) In particular, the moment conditions 𝐸𝑋=0, 𝐸𝑋²=1, and 𝐸𝑋⁴/log²|𝑋|<∞ imply that limsup_𝑛→∞∑^𝑛_𝑘=1𝑋_𝑛𝑘2𝑛log𝑛=1𝑎.𝑠. (21)

Remark 7. Formula (21) is called the law of single logarithm which is due to Hu and Weber [16]. They proved it under the strong moment condition 𝐸𝑋⁴<∞. Qi [17] and Li et al. [18] independently proved that (21) is equivalent to conditions 𝐸𝑋=0, 𝐸𝑋²=1, and 𝐸𝑋⁴/log²|𝑋|<∞. In particular, Li et al. [18] gave a version of random elements taking values in a Banach space.

Theorem 8. Let 0<𝛼<1/2, 𝑔(𝑥) be an increasing and regular varying function at infinity with index 𝛼 and let ℎ(𝑥) be the inverse function of 𝑔(𝑥). Let {𝑋,𝑋_𝑛≥1} be a sequence of independent and identically distributed random variables with 𝐸𝑋=0, 𝐸𝑋²=1, and 𝐸ℎ(|𝑋|)<∞. Suppose that {𝑎_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} is a sequence of constants such that (11) holds for some constant 𝐾>0 and limsup_𝑛→∞log𝑛^𝑛_𝑘=1𝑎²_𝑛𝑘=𝜌 (22) for some constant 0≤𝜌<∞. Then limsup_𝑛→∞^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘≤2𝜌𝑎.𝑠. (23)

Remark 9. For any 𝑝∈(0,1), set 𝑎_𝑛𝑘=(4𝑛𝜋)^1/4log𝑛^−1/2𝑛𝑘𝑝^𝑘1−𝑝^𝑛−𝑘. (24) Then, by Embrechts and Maejima [19], max_{1≤𝑘≤𝑛}𝑎_𝑛𝑘∼1𝜋^1/4𝑝(1−𝑝)^1/2⋅1𝑛^1/4log𝑛^1/2,log𝑛^𝑛_𝑘=1𝑎²_𝑛𝑘∼2⁻¹𝑝(1−𝑝)^−1/2 (25) as 𝑛→∞. Hence, from Theorem 8, 𝐸𝑋=0,𝐸𝑋²=1,𝐸(|𝑋|²/log|𝑋|)²<∞ imply that limsup_𝑛→∞(4𝑛𝜋)^1/4log𝑛^−1/2𝑛_𝑘=1𝑛𝑘𝑝^𝑘1−𝑝^𝑛−𝑘𝑋_𝑘≤𝑝1−𝑝^−1/4a.s. (26) In fact, Lai [20] has proved that limsup_𝑛→∞(4𝑛𝜋)^1/4log𝑛^−1/2𝑛_𝑘=1𝑛𝑘𝑝^𝑘1−𝑝^𝑛−𝑘𝑋_𝑘=𝑝1−𝑝^−1/4a.s. (27) if and only if 𝐸𝑋=0,𝐸𝑋²=1,𝐸(|𝑋|²/log|𝑋|)²<∞. Therefore, both the upper bound and the moment condition of Theorem 8 are sharp.

Remark 10. Let 𝛼,𝛽>0 and 1/𝛼+1/𝛽=1/2. Let {𝑏_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} be a sequence of constants with limsup_𝑛→∞𝑛⁻¹∑^𝑛_𝑘=1|𝑏_𝑛𝑘|^𝛼<∞. Set 𝑎_𝑛𝑘=(𝑛log𝑛)⁻¹𝑏_𝑛𝑘. Then, it is easy to show that max_{1≤𝑘≤𝑛}𝑎_𝑛𝑘≤𝐾𝑛^1/𝛽log𝑛⁻¹ (28) for some 𝐾>0 and 𝜌=limsup_𝑛→∞log𝑛^𝑛_𝑘=1𝑎_𝑛𝑘²=limsup_𝑛→∞𝑛^𝑛−1_𝑘=1𝑏_𝑛𝑘²≤limsup_𝑛→∞𝑛^𝑛−1_𝑘=1|𝑏_𝑛𝑘|^𝛼2/𝛼<∞ (29) by Hölder’s inequality. Hence, from Theorem 8, 𝐸𝑋=0,𝐸𝑋²=1, and 𝐸(|𝑋|^𝛽/(log|𝑋|)^𝛽/2)<∞ imply that limsup_𝑛→∞∑^𝑛_𝑘=1𝑏_𝑛𝑘𝑋_𝑘𝑛log𝑛=limsup_𝑛→∞^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘≤2𝜌a.s. (30) So, Theorem 1.4 in Chen and Gan [21] is a corollary of Theorem 8.

Remark 11. When 𝛼=1/2, Theorem 8 also holds, but the proof is completely different. So, we will discuss it in the other paper.

2. Proofs of the Main Results

The main idea in the proofs of the main results is to use the following invariance principle (see Sakhanenko [22–24]), which is a powerful tool in the field of limit theory (e.g., see Csörgő et al. [25], Jiang and Zhang [26], Chen and Gan [21], and Chen and Wang [27]).

Lemma 12. Let {𝜉_𝑘,1≤𝑘≤𝑛} be a sequence of independent random variables with 𝐸𝜉_𝑘=0 and 𝐸𝜉²_𝑘<∞ for 1≤𝑘≤𝑛. Then, there exists a sequence of independent normal random variables {𝜂_𝑘,1≤𝑘≤𝑛} with 𝐸𝜂_𝑘=0 and 𝐸𝜂²_𝑘=𝐸𝜉²_𝑘 such that, for all 𝑞>2 and 𝑦>0, 𝑃^𝑛_𝑘=1𝜉_𝑘−^𝑛_𝑘=1𝜂_𝑘>𝑦≤𝐵𝑦^𝑛−𝑞_𝑘=1𝐸𝜉_𝑘^𝑞, (31) where 𝐵 is a positive constant depending only on 𝑞.

Proof of Theorem 1. For 1≤𝑘≤𝑛 and 𝑛≥1, we let 𝑌_𝑛𝑘=𝑋_𝑘𝐼𝑋_𝑘>𝑔(𝑛)−𝐸𝑋_𝑘𝐼𝑋_𝑘(𝑛),𝑍>𝑔_𝑛𝑘=𝑋_𝑘−𝑌_𝑛𝑘. (32) We first prove that ^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘>𝜀<∞,∀𝜀>2𝑟𝜌. (33) For 𝜀>2𝑟𝜌, let 𝜀=𝜀₁+𝜀₂, where 𝜀₁>0 and 𝜀₂>2𝑟𝜌. Then ^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘≤>𝜀_∞𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑌_𝑛𝑘>𝜀₁+^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑍_𝑛𝑘>𝜀₂∶=𝐼₁+𝐼₂. (34) For 𝐼₁, we have, by Markov’s inequality, (12), (15), and a standard computation, that 𝐼₁≤𝑐^∞_𝑛=1𝑛^𝑛𝑟−1_𝑘=1𝑎²_𝑛𝑘𝐸𝑋²𝐼|𝑋|>𝑔(𝑛)≤𝑐^∞_𝑛=1𝑛^𝑟−1log𝑛𝐸𝑋²𝐼|𝑋|(𝑛)ℎ>𝑔≤𝑐𝐸^𝑟(|𝑋|)|𝑋|²logℎ(|𝑋|)<∞. (35) For 𝐼₂, we will use Lemma 12. By Lemma 12, there exists an array {𝜂_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} of rowwise independent normal random variables with 𝐸𝜂_𝑛𝑘=0,  𝐸𝜂²_𝑛𝑘=𝐸|𝑎_𝑛𝑘𝑍_𝑛𝑘|² such that, for all 𝑞>2 and 𝑦>0, 𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑍_𝑛𝑘−^𝑛_𝑘=1𝜂_𝑛𝑘>𝑦≤𝐵𝑦^𝑛−𝑞_𝑘=1𝑎_𝑛𝑘^𝑞𝐸|𝑋|^𝑞𝐼|𝑋|≤𝑔(𝑛). (36) For 𝜀₂>2𝑟𝜌, let 𝜀₂=𝜀₃+𝜀₄, where 𝜀₃>0 and 𝜀₄>2𝑟𝜌. Then 𝐼₂≤^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑍_𝑛𝑘−^𝑛_𝑘=1𝜂_𝑛𝑘>𝜀₃+^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝜂_𝑛𝑘>𝜀₄∶=𝐼₂₁+𝐼₂₂. (37) Take 𝑞>2 such that 𝑟−1−𝛼(𝑞−2)<−1. Then, we have, by (36), (11), (12), (16), and a standard computation, that 𝐼₂₁≤𝑐^∞_𝑛=1𝑛^𝑛𝑟−1_𝑘=1𝑎_𝑛𝑘^𝑞𝐸|𝑋|^𝑞𝐼|𝑋|≤𝑔(𝑛)≤𝑐^∞_𝑛=1𝑛^𝑟−1𝑔^{−(𝑞−2)}(𝑛)𝐸|𝑋|log𝑛^𝑞𝐼|𝑋|(𝑛ℎ≤𝑔)≤𝑐𝐸^𝑟(|𝑋|)|𝑋|²logℎ(|𝑋|)<∞. (38) We finally prove that 𝐼₂₂<∞. To do this, let 𝛿>1 be sufficiently close to 1 such that 𝜀₄>𝛿2𝑟𝜌. Then ∑^𝑛_𝑘=1𝑎²_𝑛𝑘≤𝜀²₄/(2𝑟𝛿log𝑛) for all large enough 𝑛. Let 𝑁 be a standard normal random variable. It is well known that 𝑃{|𝑁|>𝑥}∼2/𝜋𝑥⁻¹𝑒^−𝑥2/2. Since 𝐸𝑋²=1, 𝐸𝑍²_𝑛𝑘=𝐸{𝑋𝐼(|𝑋|≤𝑔(𝑛))−𝐸𝑋𝐼(|𝑋|≤𝑔(𝑛))}²≤1 for all 1≤𝑘≤𝑛 and 𝑛≥1. Hence, for all large enough 𝑛, 𝑃^𝑛_𝑘=1𝜂_𝑛𝑘>𝜀₄|𝑁|>𝜀≤𝑃⁴∑^𝑛_𝑘=1𝑎²_𝑛𝑘≤𝑐^𝑛_𝑘=1𝑎²_𝑛𝑘−𝜀exp²⁴2∑^𝑛_𝑘=1𝑎²_𝑛𝑘≤𝑐log𝑛^−1/2𝑛^−𝑟𝛿, (39) which gives 𝐼₂₂<∞.
Next, we prove that ^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘>𝜀=∞,∀0<𝜀<2𝑟𝜌. (40) For 0<𝜀<2𝑟𝜌, let 𝜀=−𝜀₅−𝜀₆+𝜀₇, where 𝜀₅>0,𝜀₆>0 and 0<𝜀₇<2𝑟𝜌. Then ^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝜂_𝑛𝑘>𝜀₇≤^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑖𝑌_𝑛𝑘>𝜀₅+^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑍_𝑛𝑘−^𝑛_𝑘=1𝜂_𝑛𝑘>𝜀₆+^∞_𝑛=1𝑛^𝑟−1𝑃^𝑛_𝑘=1𝑎_𝑛𝑘𝑋_𝑘>𝜀. (41) The first series on the right hand side converges by 𝐼₁<∞. The second series converges by 𝐼₂₁<∞. But by using 𝑃{|𝑁|>𝑥}∼2/𝜋𝑥⁻¹𝑒^−𝑥2/2, it is easy to show that the series on the left-hand side diverges. Hence, the last series on the right-hand side also diverges. That is, (40) holds.

Proof of Theorem 8. Let 𝛿>0 be given. Let 𝑋^′_𝑛𝑘=𝑋_𝑘𝐼𝑎_𝑛𝑘𝑋_𝑘≤𝑛^−Δ (42) for Δ>0 to be specified below and let 𝑇^′_𝑛=^𝑛_𝑘=1𝑎_𝑛𝑘𝑋^′_𝑛𝑘. (43) Let 𝑋^′′_𝑛𝑘=𝑋_𝑘𝐼𝑎_𝑛𝑘𝑋_𝑘>𝛿𝑁 (44) for an integer 𝑁 to be specified below and let 𝑇_𝑛^′′=^𝑛_𝑘=1𝑎_𝑛𝑘𝑋^′′_𝑛𝑘. (45) Let 𝑋^′′′_𝑛𝑘=𝑋_𝑘−𝑋^′_𝑛𝑘−𝑋^′′_𝑛𝑘=𝑋_𝑘𝐼𝑛^−Δ<𝑎_𝑛𝑘𝑋_𝑘≤𝛿𝑁 (46) and let 𝑇_𝑛^′′′=^𝑛_𝑘=1𝑎_𝑛𝑘𝑋^′′′_𝑛𝑘. (47) Since 𝛿>0 is arbitrary, to prove (22), it is enough to show that limsup_𝑛→∞𝑇^′_𝑛−𝐸𝑇^′_𝑛≤2𝜌a.s.𝐸𝑇^′_𝑛𝑇⟶0_𝑛′′⟶0a.s.limsup_𝑛→∞𝑇_𝑛^′′′≤𝛿a.s. (48) Using the same argument as in the proof of Theorem 1, we have that, for any 𝜀>2𝜌 with 𝜀=𝜀₁+𝜀₂, where 𝜀₁>2𝜌 and 𝜀₂>0, ^∞_𝑛=1𝑃𝑇^′_𝑛−𝐸𝑇^′_𝑛≤>𝜀_∞𝑛=1𝑃^𝑘_𝑘=1𝜂_𝑛𝑘>𝜀₁+𝑃𝑇^′_𝑛−𝐸𝑇^′_𝑛−^𝑘_𝑘=1𝜂_𝑛𝑘>𝜀₂≤𝑐^∞_𝑛=1^𝑛_𝑘=1𝑎²_𝑛𝑘−𝜀exp²¹2∑^𝑛_𝑘=1𝑎²_𝑛𝑘+𝑐^∞𝑛_𝑛=1𝑘=1𝐸𝑎_𝑛𝑘𝑋^′_𝑛𝑘^𝑞≤𝑐^∞_𝑛=1−𝜀exp²¹2∑^𝑛_𝑘=1𝑎²_𝑛𝑘+𝑐^∞_𝑛=1𝑛^1−𝑞Δ<∞ (49) when 𝑞>2 is large enough. Hence, by Borel-Cantelli lemma, limsup_𝑛→∞|𝑇^′_𝑛−𝐸𝑇^′_𝑛|≤2𝜌a.s. Taking 𝑟∈(2,1/𝛼), we get, by 𝐸𝑋=0, (11), and (22), that 𝐸𝑇^′_𝑛≤^𝑛_𝑘=1𝐸𝑎_𝑛𝑘𝑋_𝑘𝐼𝑎_𝑛𝑘𝑋_𝑘>𝑛^−Δ≤𝑛^{𝑛(𝑟−1)Δ}_𝑘=1𝐸𝑎_𝑛𝑘𝑋_𝑘^𝑟𝐼𝑎_𝑛𝑘𝑋_𝑘>𝑛^−Δ≤𝑐𝑛^{𝑛(𝑟−1)Δ}_𝑘=1𝑎_𝑛𝑘^𝑟≤𝑐𝑛^(𝑟−1)Δ𝑔(𝑛)^{−(𝑟−2)}; (50) Hence, 𝐸𝑇^′_𝑛→0 when Δ is small enough. By 𝐸ℎ(|𝑋|)<∞, ^∞_𝑘=1𝑃𝑋|>𝛿|𝑔(𝑘𝑁𝐾)<∞ (51) which implies that the series ∑^∞_𝑘=1|𝑋_𝑘|𝐼(|𝑋_𝑘|>(𝛿/𝑁𝐾)𝑔(𝑘)) converges almost surely by Borel-Cantelli lemma. Hence, 𝑇_𝑛^′′≤𝐾𝑔⁻¹(𝑛)^𝑛_𝑘=1𝑋_𝑘𝐼𝑋_𝑘>𝛿𝑔(𝑛𝑁𝐾)≤𝐾𝑔⁻¹(𝑛)^∞_𝑘=1𝑋_𝑘𝐼𝑋_𝑘>𝛿𝑔(𝑘)𝑁𝐾⟶0a.s. (52) To prove limsup_𝑛→∞|𝑇_𝑛^′′′|≤𝛿a.s., by Borel-Cantelli lemma, it is enough to show that ^∞_𝑛=1𝑃𝑇_𝑛^′′′>𝛿<∞. (53) Since 𝑋^′′′_𝑛𝑘=𝑋_𝑘𝐼(𝑛^−Δ<|𝑎_𝑛𝑘𝑋_𝑘|≤𝛿/𝑁), |𝑇_𝑛^′′′|=|∑^𝑛_𝑘=1𝑎_𝑛𝑘𝑋^′′′_𝑛𝑘|>𝛿 implies that there must exist at least 𝑁 indices 𝑘 such that 𝑋^′′′_𝑛𝑘≠0. Hence, by Bonferroni’s inequality (see, e.g., Lemma  4.1.2 in [1]) and Markov’s inequality, 𝑃𝑇_𝑛^′′′>𝛿≤𝑃thereexistatleast𝑁indices𝑘suchthat𝑋^′′′_𝑛𝑘≤≠0_{1≤𝑘1<𝑘2<⋯<𝑘𝑁𝑁≤𝑛𝑗=1}𝑃𝑎_𝑛𝑘𝑗𝑋_𝑘𝑗>𝑛^−Δ≤^𝑛_𝑘=1𝑃𝑎_𝑛𝑘𝑋_𝑘>𝑛^−Δ𝑁≤𝑛^𝑛𝑝Δ_𝑘=1𝑎_𝑛𝑘^𝑝𝐸|𝑋|^𝑝𝑁≤𝑐𝑛^𝑝Δ𝑁𝑔(𝑛)^{−𝑁(𝑝−2)}, (54) where 𝑝∈(2,1/𝛼). Thus, choosing Δ sufficiently small and 𝑁 sufficiently large such that 𝑁[𝛼(𝑝−2)−𝑝Δ]>1, we have ^∞_𝑛=1𝑃𝑇_𝑛^′′′>𝛿≤𝑐^∞_𝑛=1𝑛^𝑝Δ𝑁𝑔(𝑛)^{−𝑁(𝑝−2)}<∞, (55) since the function 𝑥^𝑝Δ𝑁[𝑔(𝑥)]^{−𝑁(𝑝−2)} is regularly varying at infinity with index −𝑁[𝛼(𝑝−2)−𝑝Δ]<−1. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for the helpful comments. The research of Pingyan Chen is supported by the National Natural Science Foundation of China (11271161). The research of Soo Hak Sung is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013131).