1.Introduction YuanyingJiang andQunyingWu TheAlmostSureLocalCentralLimitTheoremfortheNegativelyAssociatedSequences ResearchArticle

Volume 2013, Article ID 656257,9pages http://dx.doi.org/10.1155/2013/656257

Research Article

The Almost Sure Local Central Limit Theorem for the Negatively Associated Sequences

Yuanying Jiang

and Qunying Wu

1College of Science, Guilin University of Technology, Guilin 541004, China

2School of Statistics, Renmin University of China, Beijing 100872, China

Correspondence should be addressed to Yuanying Jiang; [email protected] Received 13 May 2013; Accepted 18 June 2013

Academic Editor: Ying Hu

Copyright © 2013 Y. Jiang and Q. Wu. 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.

In this paper, the almost sure central limit theorem is established for sequences of negatively associated random variables:

lim_{𝑛 → ∞}(1/log𝑛) ∑^𝑛_𝑘=1(𝐼(𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘)/𝑘)𝑃(𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘) = 1, almost surely. This is the local almost sure central limit theorem for negatively associated sequences similar to results by Cs´aki et al. (1993). The results extend those on almost sure local central limit theorems from the i.i.d. case to the stationary negatively associated sequences.

1. Introduction

Definition 1. Random variables 𝑋₁, 𝑋₂, . . . , 𝑋_𝑛, 𝑛 ≥ 2are said to be negatively associated (NA) if for every pair of disjoint subsets𝐴₁and𝐴₂of{1, 2, . . . , 𝑛},

Cov(𝑓₁(𝑋_𝑖, 𝑖 ∈ 𝐴₁) , 𝑓₂(𝑋_𝑗, 𝑗 ∈ 𝐴₂)) ≤ 0, (1) where𝑓₁and𝑓₂are increasing for every variable (or decreas- ing for every variable) such that this covariance exists. A sequence of random variables{𝑋_𝑖, 𝑖 ≥ 1}is said to be NA if every finite subfamily of{𝑋_𝑖, 𝑖 ≥ 1}is NA.

Obviously, if {𝑋_𝑖, 𝑖 ≥ 1} is a sequence of NA random variables and {𝑓_𝑖, 𝑖 ≥ 1} is a sequence of nondecreasing (or nonincreasing) functions, then{𝑓_𝑖(𝑋_𝑖), 𝑖 ≥ 1}is also a sequence of NA random variables.

This definition was introduced by the Joag-Dev and Proschan [1]. Statistical test depends greatly on sampling, and the random sampling without replacement from a finite population is NA, but it is not independent. NA sampling has wide applications such as those in multivariate statistical analysis and reliability theory. Because of the wide applica- tions of NA sampling, the notions of NA random variables have received more and more attention recently. We refer to Joag-Dev and Proschan [1] for fundamental properties, Shao

[2] for the moment inequalities, and Wu and Jiang [3] for Chover’s law of the iterated logarithm.

Assume that{𝑋_𝑛, 𝑛 ≥ 1}is a strictly stationary sequence of NA random variables with𝐸𝑋₁= 0, 0 < 𝐸𝑋²₁< ∞. Define 𝑆_𝑛= ∑^𝑛_𝑗=1𝑋_𝑗,

𝜎²_𝑛:=Var𝑆_𝑛, (2)

𝜎²:=Var𝑋₁+ 2∑^∞

𝑗=2

Cov(𝑋₁, 𝑋_𝑗) . (3)

(1) Newman [4] and Matuła [5] showed that NA station- ary sequences satisfy the central limit theorem (CLT) under𝜎²> 0, that is,

sup𝑥∈𝑅

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑃(𝑆_𝑛

𝜎_𝑛 < 𝑥) − Φ (𝑥)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = 𝑜(1). (4) (2) Applying Matuła [6] and Wu’s [7] methods, we can easily show that NA sequences satisfy the almost sure central limit theorem (ASCLT), that is,

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

1

𝑘𝐼 { 𝑆_𝑘 ≤ 𝑥𝜎𝑘^1/2} = Φ (𝑥) a.s. ∀𝑥 ∈R, (5) whereΦ(𝑥)is the standard normal distribution func- tion and𝐼{𝐴}denotes the indicator of the event𝐴.

The ASCLT was stimulated by Brosamler [8] and Schatte [9]. Both were concerned with the partial sum of independent and identically distributed (i.i.d.) random variables with more than the second moment. The ASCLT was extensively studied in the past two decades and an interesting direction of the study is to prove it for dependent variables. There are some results for weakly dependent variables such as𝛼, 𝜌, 𝜙-mixing and associated random variables. Among those results, we refer to Peligrad and Shao [10], Matuła [11], and Wu [7].

More general version of ASCLT was proved by Cs´aki et al.

[12]. The following theorem is due to them.

Theorem A. Let {𝑋_𝑛, 𝑛 ≥ 1}be a sequence of i.i.d. random variables with𝐸|𝑋₁|³< ∞, let𝐸𝑋₁= 0.𝑎_𝑘, 𝑏_𝑘satisfy

−∞ ≤ 𝑎_𝑘≤ 0 ≤ 𝑏_𝑘≤ ∞, 𝑘 = 1, 2, . . . , (6) and assume that

∑𝑛 𝑘=1

log𝑘

𝑘^3/2𝑃 (𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘) = 𝑂 (log𝑛) , 𝑎𝑠 𝑛 󳨀→ ∞, (7) and then

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

𝐼 {𝑎_𝑘≤ 𝑆_𝑘 < 𝑏_𝑘}

𝑘𝑃 (𝑎_𝑘 ≤ 𝑆_𝑘< 𝑏_𝑘)= 1 a.s. (8) This result may be called almost sure local central limit theorem, while (5) may be called almost sure global central limit theorem. Hurelbaatar [13] extended (8) to the case of𝜌- mixing sequences and Weng et al. [14] derived an almost sure local central limit theorem for the product of partial sums of a sequence of i.i.d. positive random variables under some regular conditions. For more details, we refer to Berkes and Cs´aki [15] and F¨oldes [16].

Our concern in this paper is to give a common general- ization of (8) to the case of NA sequences.

In the next section we present the exact results, postpon- ing some technical lemmas and the proofs toSection 3.

2. Main Results

Assume in the following that {𝑋_𝑛, 𝑛 ≥ 1} is a strictly stationary sequence of NA random variables with𝐸𝑋₁ = 0, 0 < 𝐸𝑋²₁ < ∞. We consider the limit behavior of the logarithmic average

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

𝐼 {𝑎_𝑘 ≤ 𝑆_𝑘< 𝑏_𝑘}

𝑘𝑃 (𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘) (9) with−∞ ≤ 𝑎_𝑘 ≤ 0 ≤ 𝑏_𝑘 ≤ ∞, where the terms in the sum above are defined to be1if their denominator happens to be 0.

More precisely let {𝑎_𝑛, 𝑛 ≥ 1} and{𝑏_𝑛, 𝑛 ≥ 1} be two sequences of real numbers and put

𝑝_𝑘:= 𝑃 (𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘) ,

𝛼_𝑘:={{ {{ {

𝐼 {𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘}

𝑝_𝑘 , if 𝑝_𝑘 ̸= 0,

1, if 𝑝_𝑘= 0.

(10)

So we need to investigate the limit behavior of 𝜇_𝑛:=∑^𝑛

𝑘=1

𝛼_𝑘

𝑘 (11)

under certain conditions.

In our considerations, we will need the following Cox- Grimmett coefficient which describes the covariance struc- ture of the sequence

𝑢 (𝑛) := sup

𝑘∈𝑁 ∑

𝑗:|𝑗−𝑘|≥𝑛󵄨󵄨󵄨󵄨󵄨Cov(𝑋_𝑗, 𝑋_𝑘)󵄨󵄨󵄨󵄨󵄨, 𝑛 ∈ 𝑁 ∪ {0}. (12) We remark that for a stationary sequence of NA random variables

𝑢 (𝑛) = −2 ∑^∞

𝑘=𝑛+1

Cov(𝑋₁, 𝑋_𝑘) , 𝑛 ∈ 𝑁. (13) By Lemma 8 of Newman [4], we have 𝑢(0) < ∞ and lim_{𝑛 → ∞}𝑢(𝑛) = 0.

In the following,𝜉_𝑛 ∼ 𝜂_𝑛denotes𝜉_𝑛/𝜂_𝑛 → 1, 𝑛 → ∞.

𝜉_𝑛 = 𝑂(𝜂_𝑛)denotes that there exists a constant𝑐 > 0such that𝜉_𝑛 ≤ 𝑐𝜂_𝑛for sufficiently large𝑛. The symbols𝑐, 𝑐₁, 𝑐₂, . . ., stand for generic positive constants which may differ from one place to another.

Theorem 2. Let{𝑋_𝑛, 𝑛 ≥ 1}be a strictly stationary sequence of NA random variables with𝐸𝑋₁ = 0, 𝐸|𝑋₁|³ < ∞and let 𝜎²> 0. 𝑎_𝑘, 𝑏_𝑘satisfy(6). Assume that

∑∞

𝑛=1𝑢 (𝑛) < ∞, (14)

and for some𝛽 > 1,

∑

1≤𝑘≤𝑛 𝑝𝑘 ̸= 0

(log𝑘)^1/3

𝑘𝑝_𝑘 = 𝑂 ((log𝑛)²(log log𝑛)^−𝛽) . (15)

Then we have

𝑛 → ∞lim 𝜇_𝑛

log𝑛= 1, a.s., (16)

where𝜇_𝑛is defined by(11).

Remark 3. Let𝑎_𝑘 = −∞and𝑏_𝑘= 𝑥𝜎𝑘^1/2in (6). By the central limit theorem (4), we have𝑝_𝑘 = 𝑃(𝑆_𝑘/𝜎𝑘^1/2 < 𝑥) → Φ(𝑥), obviously (15) satisfies; then (16) becomes (5), which is the almost sure global central limit theorem. Thus the almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem.

Remark 4. The condition (15) is satisfied with a wide range of 𝑝_𝑘; for example, if

𝑝_𝑘= 0 or 𝑝_𝑘≥ 𝑐(log log𝑘)^𝛽

(log𝑘)^2/3 (17)

holds, then the condition (15) is satisfied. In fact, letting0 <

𝛿 < 1, we have

∑

1≤𝑘≤𝑛 𝑝𝑘 ̸= 0

(log𝑘)^1/3 𝑘𝑝_𝑘

≤ 𝑐 ∑

1≤𝑘≤𝑛

(log log𝑘)^−𝛽(log𝑘)^𝛿(log𝑘)^1−𝛿 𝑘

≤ 𝑐(log log𝑛)^−𝛽(log𝑛)^𝛿 ∑

1≤𝑘≤𝑛

(log𝑘)^1−𝛿 𝑘

≤ 𝑐(log log𝑛)^−𝛽(log𝑛)²

= 𝑂 ((log𝑛)²(log log𝑛)^−𝛽) .

(18)

In the given theorem below, we strengthen the condition (6) on𝑎_𝑘and𝑏_𝑘. Meanwhile, as a compensation, we do not need to impose restricting condition (15) on𝑝_𝑘.

Theorem 5. Let{𝑋_𝑛, 𝑛 ≥ 1}be a strictly stationary sequence of NA random variables with𝐸𝑋₁= 0, 𝐸|𝑋₁|³< ∞, and𝜎²> 0, and let𝑎_𝑘and𝑏_𝑘satisfy

−𝑐₁𝑘^1/2−𝛼≤ 𝑎_𝑘≤ −𝑐₂𝑘^1/2−𝛼,

𝑐₃𝑘^1/2−𝛼≤ 𝑏_𝑘 ≤ 𝑐₄𝑘^1/2−𝛼, (19) where0 < 𝛼 < 1/7. Assume that(14)hold, and then we have (16).

3. Proofs

The following lemmas play important roles in the proof of our theorems. The proofs are given in the Appendix.

Lemma 6. Assume that{𝜉_𝑛, 𝑛 ≥ 1}are random variables such that

𝜉_𝑘 ≥ 0, 𝐸𝜉_𝑘= 1, 𝑘 = 1, 2, . . . (20) and furthermore there exists𝑑_𝑘 ≥ 0such that𝐷_𝑛= ∑^𝑛_𝑘=1𝑑_𝑘↑

∞, 𝐷_𝑛/𝐷_𝑛−1 → 1, and Var(∑^𝑛

𝑘=1

𝑑_𝑘𝜉_𝑘) ≤ 𝑐𝐷²_𝑛(log𝐷_𝑛)^−𝛽, (21)

with some𝛽 > 1and positive constant𝑐, and then

𝑛 → ∞lim 1 𝐷_𝑛

∑𝑛 𝑘=1

𝑑_𝑘𝜉_𝑘= 1 a.s. (22) Remark 7. Let𝑑_𝑘 = 1/𝑘in (21), and then𝐷_𝑛 = ∑^𝑛_𝑘=11/𝑘 ∼ log𝑛. Thus, if

Var(∑^𝑛

𝑘=1

1

𝑘𝜉_𝑘) ≤ 𝑐(log𝑛)²(log log𝑛)^−𝛽, (23)

with some𝛽 > 1and positive constant𝑐, then

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

1

𝑘𝜉_𝑘= 1 a.s. (24) The followingLemma 8is obvious.

Lemma 8. Assume that the nonnegative random sequence {𝜉_𝑛, 𝑛 ≥ 1}satisfies(24)and the sequence{𝜂_𝑛, 𝑛 ≥ 1}is such that, for any𝜀 > 0, there exists a𝑘₀= 𝑘₀(𝜀, 𝜔)for which

(1 − 𝜀) 𝜉_𝑘 ≤ 𝜂_𝑘≤ (1 + 𝜀) 𝜉𝑘, 𝑘 > 𝑘₀. (25) Then we have also

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

1

𝑘𝜂_𝑘 = 1 a.s. (26) The followingLemma 9is an easy corollary to the Corol- lary 2.2 in Matuła [5] under strictly stationary condition, which studies the rate of convergence in the CLT under neg- ative dependence. It was also studied in Pan [17]. Of course this is the Berry-Esseen inequality for the NA sequence.

Lemma 9. Let{𝑋_𝑗, 𝑗 ∈ 𝑁}be a strictly stationary sequence of NA random variables with𝐸𝑋₁ = 0, 𝐸|𝑋₁|³ < ∞, 𝜎² > 0 satisfying(14). Then one has

sup𝑥∈𝑅

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑃(𝑆_𝑛

𝜎_𝑛 < 𝑥) − Φ (𝑥)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨 = 𝑂(𝑛^−1/5) . (27) Lemma 10. If the conditions ofLemma 9hold,𝑎_𝑘and𝑏_𝑘satisfy (19). Then one has

𝑐₁^󸀠𝑘^−𝛼 ≤ 𝑝_𝑘≤ 𝑐₂^󸀠𝑘^−𝛼 (𝑘 ≥ 𝑘₀) , (28) where𝛼is as(19).

Lemma 11. If the conditions ofLemma 9hold,𝑎_𝑘and𝑏_𝑘satisfy (19), and 𝛼is as(19). Assume that𝜀_𝑙 = 𝑙^3𝛼/2, and then the following asymptotic relations hold:

∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1

𝑘𝑙𝑝_𝑙𝑃 (󵄨󵄨󵄨󵄨𝑆^𝑙𝛼󵄨󵄨󵄨󵄨 ≥ 𝜀^𝑙) = 𝑂 (log𝑛) , (29)

∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1 𝑘𝑙𝑝_𝑙

1

(𝑙 − 𝑘 − 𝑙^𝛼)^1/5 = 𝑂 (log𝑛) , (30)

∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1

𝑘𝑙𝑝_𝑙󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨Φ ( 𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑙

(𝑙 − 𝑘 − 𝑙^𝛼)^1/2) − Φ ( 𝑎_𝑙 𝑙^1/2)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

= 𝑂 (log𝑛) ,

(31)

∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1

𝑘𝑙𝑝_𝑙󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(Φ ( 𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑙

(𝑙 − 𝑘 − 𝑙^𝛼)^1/2) − Φ ( 𝑏_𝑙 𝑙^1/2))󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

= 𝑂 (log𝑛) .

(32)

The main point in our proof is to verify the condition (23).

We use global central limit theorem with remainders and the following elementary inequalities:

󵄨󵄨󵄨󵄨Φ(𝑥) − Φ(𝑦)󵄨󵄨󵄨󵄨 ≤ 𝑐󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨, for every𝑥, 𝑦 ∈R (33) with some constant𝑐. Moreover, for each𝑘 > 0, there exists 𝑐₁= 𝑐₁(𝑘), such that

󵄨󵄨󵄨󵄨Φ(𝑥) − Φ(𝑦)󵄨󵄨󵄨󵄨

≥ 𝑐₁󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨, for every𝑥, 𝑦 ∈R, |𝑥| + 󵄨󵄨󵄨󵄨𝑦󵄨󵄨󵄨󵄨 ≤ 𝑘. (34) Let{𝑋_𝑛, 𝑛 ≥ 1}be a strictly stationary sequence of NA random variables with𝜎² > 0; we can immediately get𝜎_𝑛² ∼ 𝑛𝜎², that is,

𝑐₁𝑛 ≤Var(𝑆_𝑛) = 𝜎_𝑛²≤ 𝑐₂𝑛 (35) for some constant𝑐₁, 𝑐₂> 0and sufficiently large𝑛.

Proof ofTheorem 2. First assume that

𝑏_𝑘− 𝑎_𝑘≤ 𝑐𝑘^1/2, 𝑘 = 1, 2, . . . (36) with some constant𝑐. Let1 ≤ 𝑘 < 𝑙and𝜀_𝑘 = 𝑘^1/2(log𝑘)^1/3.

If either𝑝_𝑘 = 0or𝑝_𝑙= 0, then obviously Cov(𝛼_𝑘, 𝛼_𝑙) = 0, and so we may assume that𝑝_𝑘𝑝_𝑙 ̸= 0; then, we have

Cov(𝛼_𝑘, 𝛼_𝑙)

= 1

𝑝_𝑙𝑝_𝑘(𝑃 (𝑎_𝑘≤ 𝑆_𝑘 < 𝑏_𝑘, 𝑎_𝑙≤ 𝑆_𝑙< 𝑏_𝑙) − 𝑝_𝑙𝑝_𝑘)

≤ 1

𝑝_𝑙𝑝_𝑘(𝑃 (𝑎_𝑙− 𝑏_𝑘≤ 𝑆_𝑙− 𝑆_𝑘< 𝑏_𝑙 − 𝑎_𝑘, 𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘) − 𝑝_𝑙𝑝_𝑘)

≤ 1

𝑝_𝑙𝑝_𝑘(𝑃 (𝑎_𝑙− 𝑏_𝑘≤ 𝑆_𝑙− 𝑆_𝑘< 𝑏_𝑙− 𝑎_𝑘)

×𝑃 (𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘) − 𝑝_𝑙𝑝_𝑘)

= 1

𝑝_𝑙(𝑃 (𝑎_𝑙− 𝑏_𝑘 ≤ 𝑆_𝑙− 𝑆_𝑘< 𝑏_𝑙− 𝑎_𝑘) − 𝑝_𝑙)

≤ 1

𝑝_𝑙(𝑃 (𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑘< 𝑆_𝑙< 𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑘)

−𝑝_𝑙+ 𝑃 (󵄨󵄨󵄨󵄨𝑆^𝑘󵄨󵄨󵄨󵄨 ≥ 𝜀^𝑘))

≤ 1

𝑝_𝑙(𝑃 (𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑘≤ 𝑆_𝑙< 𝑎_l)

+ 𝑃 (𝑏_𝑙≤ 𝑆_𝑙< 𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑘) +𝑃 (󵄨󵄨󵄨󵄨𝑆^𝑘󵄨󵄨󵄨󵄨 ≥ 𝜀^𝑘)) . (37)

ApplyingLemma 9, (33), (35), and (36) and noting that 𝜀_𝑘= 𝑘^1/2(log𝑘)^1/3, we obtain

𝑃 (𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑘 ≤ 𝑆_𝑙< 𝑎_𝑙) + 𝑃 (𝑏_𝑙≤ 𝑆_𝑙< 𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑘)

≤ (Φ (𝑎_𝑙

𝜎_𝑙) − Φ (𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑘 𝜎_𝑙 )) + (Φ (𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑘

𝜎_𝑙 ) − Φ (𝑏_𝑙

𝜎_𝑙)) + 𝑐 1 𝑙^1/5

≤ 𝑏_𝑘+ 𝜀_𝑘

𝜎_𝑙 +−𝑎_𝑘+ 𝜀_𝑘 𝜎_𝑙 + 𝑐 1

𝑙^1/5 = 𝑏_𝑘− 𝑎_𝑘 𝜎_𝑙 +2𝜀_𝑘

𝜎_𝑙 + 𝑐 1 𝑙^1/5

≤ 𝑐 (𝑘^1/2 𝑙^1/2 + 𝜀_𝑘

𝑙^1/2 + 1

𝑙^1/5) ≤ 𝑐 (𝑘^1/2(log𝑘)^1/3 𝑙^1/2 + 1

𝑙^1/5) . (38) By the condition of (15), we have

∑𝑛 𝑙=1

∑𝑙 𝑘=1

1 𝑘𝑙𝑝_𝑙

𝑘^1/2(log𝑘)^1/3 𝑙^1/2

≤∑^𝑛

𝑙=1

1 𝑙^3/2𝑝_𝑙

∑𝑙 𝑘=1

(log𝑘)^1/3 𝑘^1/2

≤ 𝑐∑^𝑛

𝑙=1

(log𝑙)^1/3 𝑙^3/2𝑝_𝑙

∑𝑙 𝑘=1

1 𝑘^1/2 ≤ 𝑐∑^𝑛

𝑙=1

(log𝑙)^1/3 𝑙^3/2𝑝_𝑙 𝑙^1/2

≤ 𝑐∑^𝑛

𝑙=1

(log𝑙)^1/3

𝑙𝑝_𝑙 = 𝑂 ((log𝑛)²(log log𝑛)^−𝛽) ,

∑𝑛 𝑙=1

𝑙−1∑

𝑘=1

1 𝑘𝑙𝑝_𝑙

1 𝑙^1/5

=∑^𝑛

𝑙=1

1 𝑙^6/5𝑝_𝑙

𝑙−1∑

𝑘=1

1 𝑘 ≤ 𝑐∑^𝑛

𝑙=1

1 𝑙^6/5𝑝_𝑙log𝑙

≤ 𝑐∑^𝑛

𝑙=1

(log𝑙)^1/3

𝑙𝑝_𝑙 = 𝑂 ((log𝑛)²(log log𝑛)^−𝛽) . (39)

By Chebyshev’s inequality and the condition of (15) and (35), we obtain

∑𝑛 𝑙=1

∑𝑙 𝑘=1

1

𝑘𝑙𝑝_𝑙𝑃 (󵄨󵄨󵄨󵄨𝑆^𝑘󵄨󵄨󵄨󵄨 ≥ 𝜀^𝑘)

≤∑^𝑛

𝑙=1

1 𝑙𝑝_𝑙

∑𝑙 𝑘=1

Var(𝑆_𝑘) 𝑘𝜀²_𝑘 ≤ 𝑐∑^𝑛

𝑙=1

1 𝑙𝑝_𝑙

∑𝑙 𝑘=1

1 𝑘(log𝑘)^2/3

≤ 𝑐∑^𝑛

𝑙=1

(log𝑙)^1/3

𝑙𝑝_𝑙 = 𝑂 ((log𝑛)²(log log𝑛)^−𝛽) . (40)

Hence (37)–(40) imply that

∑𝑛 𝑙=1

∑𝑙−1 𝑘=1

Cov(𝛼_𝑘, 𝛼_𝑙)

𝑘𝑙 = 𝑂 ((log𝑛)²(log log𝑛)^−𝛽) . (41)

But Var(𝛼_𝑘) = 0if𝑝_𝑘= 0and Var(𝛼_𝑘) =1 − 𝑝_𝑘

𝑝_𝑘 ≤ 1

𝑝_𝑘 if𝑝_𝑘 ̸= 0. (42) Thus

∑𝑛 𝑘=1

Var(𝛼_𝑘) 𝑘²

≤ ∑

1≤𝑘≤𝑛 𝑝𝑘 ̸= 0

1

𝑘²𝑝_𝑘 ≤ ∑

1≤𝑘≤𝑛 𝑝𝑘 ̸= 0

(log𝑘)^1/3 𝑘𝑝_𝑘

= 𝑂 ((log𝑛)²(log log𝑛)^−𝛽) .

(43)

Equations (41) and (43) together imply that Var(∑^𝑛

𝑘=1

𝛼_𝑘

𝑘) = 𝑂 ((log𝑛)²(log log𝑛)^−𝛽) , as𝑛 → ∞.

(44) Hence applyingRemark 7, our theorem is proved under the restricting condition (36).

Now we drop the restricting condition (36). Fix𝑥 > 0and define

̃𝑎_𝑘=max(𝑎_𝑘, −𝑥𝜎𝑘^1/2) ,

̃𝑏_𝑘=min(𝑏_𝑘, 𝑥𝜎𝑘^1/2) ,

̃𝑝_𝑘 = 𝑃 (̃𝑎_𝑘≤ 𝑆_𝑘< ̃𝑏_𝑘) ,

(45)

where𝜎is defined by (3).

Clearly ̃𝑝_𝑘 ≤ 𝑝_𝑘, and so assuming ̃𝑝_𝑘 ̸= 0; then, also we have𝑝_𝑘 ̸= 0, and thus

𝛼_𝑘= 1

𝑝_𝑘𝐼 {𝑎_𝑘 ≤ 𝑆_𝑘 < 𝑏_𝑘}

= 1

𝑝_𝑘(𝐼 {̃𝑎_𝑘≤ 𝑆_𝑘 < ̃𝑏_𝑘}

+𝐼 {𝑎_𝑘≤ 𝑆_𝑘< ̃𝑎_𝑘} + 𝐼 {̃𝑏_𝑘≤ 𝑆_𝑘< 𝑏_𝑘})

≤ 1

̃𝑝_𝑘𝐼 {̃𝑎_𝑘≤ 𝑆_𝑘 < ̃𝑏_𝑘} + 1

𝑝_𝑘 (𝐼 {𝑎_𝑘≤ 𝑆_𝑘< ̃𝑎_𝑘} + 𝐼 {̃𝑏_𝑘 ≤ 𝑆_𝑘 < 𝑏_𝑘})

≤ 1

̃𝑝_𝑘𝐼 {̃𝑎_𝑘≤ 𝑆_𝑘 < ̃𝑏_𝑘} + 𝐼 { 𝑆_𝑘< −𝑥𝜎𝑘^1/2}

𝑃 (−𝑥𝜎𝑘^1/2≤ 𝑆_𝑘< 0)+ 𝐼 { 𝑆_𝑘≥ 𝑥𝜎𝑘^1/2} 𝑃 (0 ≤ 𝑆_𝑘 < 𝑥𝜎𝑘^1/2).

(46)

By (35) and the central limit theorem for NA random variables (4), that is,

sup𝑥∈𝑅

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑃(𝑆_𝑛

𝜎_𝑛 < 𝑥) − Φ (𝑥)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = 𝑜(1), (47)

we obtain

𝑘 → ∞lim𝑃 (−𝑥𝜎𝑘^1/2≤ 𝑆_𝑘< 0)

= lim

𝑘 → ∞𝑃 (−𝑥𝜎_𝑘≤ 𝑆_𝑘< 0) = Φ (0) − Φ (−𝑥) ,

𝑘 → ∞lim𝑃 (0 ≤ 𝑆_𝑘< 𝑥𝜎𝑘^1/2)

= lim

𝑘 → ∞𝑃 (0 ≤ 𝑆_𝑘< 𝑥𝜎_𝑘) = Φ (𝑥) − Φ (0) .

(48)

Applying the almost sure central limit theorem for NA random variables (5), that is,

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

1

𝑘𝐼 {𝑆_𝑘≤ 𝑥𝜎𝑘^1/2} = Φ (𝑥) a.s. ∀𝑥 ∈R, (49) Lemma 8, and (48), we have

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

𝐼 { 𝑆_𝑘< −𝑥𝜎𝑘^1/2} 𝑘𝑃 (−𝑥𝜎𝑘^1/2≤ 𝑆_𝑘< 0)

= Φ (−𝑥)

Φ (0) − Φ (−𝑥) a.s.,

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

𝐼 { 𝑆_𝑘> 𝑥𝜎𝑘^1/2} 𝑘𝑃 (0 ≤ 𝑆_𝑘 < 𝑥𝜎𝑘^1/2)

= 1 − Φ (𝑥) Φ (𝑥) − Φ (0) a.s.

(50)

Sincẽ𝑎_𝑘and̃𝑏_𝑘satisfy (36), we get

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

̃𝛼_𝑘

𝑘 = 1 a.s., (51)

where

̃𝛼_𝑘={{ {{ {

𝐼 {̃𝑎_𝑘≤ 𝑆_𝑘< ̃𝑏_𝑘}

̃𝑝_𝑘 , if ̃𝑝_𝑘 ̸= 0,

1, if ̃𝑝_𝑘 = 0.

(52)

Equations (46) and (50)–(51) together imply that lim sup

𝑛 → ∞

1 log𝑛

∑𝑛 𝑘=1

𝛼_𝑘

𝑘 ≤ 1 + 2 1 − Φ (𝑥)

Φ (𝑥) − Φ (0) a.s. (53) On the other hand, if̃𝑝_𝑘 ̸= 0, then we have

1

𝑝_𝑘𝐼 {𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘}

≥ 1

̃𝑝_𝑘𝐼 {̃𝑎_𝑘 ≤ 𝑆_𝑘< ̃𝑏_𝑘} (1 −𝑝_𝑘− ̃𝑝_𝑘 𝑝_𝑘 )

≥ ̃𝛼_𝑘(1 − 𝑃 (𝑆_𝑘< −𝜎𝑥𝑘^1/2) + 𝑃 (𝑆_𝑘> 𝜎𝑥𝑘^1/2) min{𝑃 (0 ≤ 𝑆_𝑘 < 𝜎𝑥𝑘^1/2) , 𝑃 (−𝜎𝑥𝑘^1/2≤ 𝑆_𝑘< 0)}),

(54)

and by the central limit theorem,

𝑘 → ∞lim

𝑃 (𝑆_𝑘< −𝜎𝑥𝑘^1/2) + 𝑃 (𝑆_𝑘> 𝜎𝑥𝑘^1/2) min{𝑃 (0 ≤ 𝑆_𝑘< 𝜎𝑥𝑘^1/2) , 𝑃 (−𝜎𝑥𝑘^1/2≤ 𝑆_𝑘 < 0)}

= 1 − 2 1 − Φ (𝑥) Φ (𝑥) − Φ (0).

(55) ApplyingLemma 8, (51) and (54) imply that

lim inf_{𝑛 → ∞} 1 log𝑛

∑𝑛 𝑘=1

𝛼_𝑘

𝑘 ≥ 1 − 2 1 − Φ (𝑥)

Φ (𝑥) − Φ (0) a.s., (56) and hence

1 + 2 1 − Φ (𝑥)

Φ (𝑥) − Φ (0) ≥lim sup

𝑛 → ∞

1 log𝑛

∑𝑛 𝑘=1

𝛼_𝑘 𝑘

≥lim inf

𝑛 → ∞

1 log𝑛

∑𝑛 𝑘=1

𝛼_𝑘 𝑘

≥ 1 − 2 1 − Φ (𝑥) Φ (𝑥) − Φ (0) a.s.

(57)

By the arbitrariness of𝑥, let𝑥 → ∞in (57); we have 1 ≥lim sup

𝑛 → ∞

1 log𝑛

∑𝑛 𝑘=1

𝛼_𝑘 𝑘

≥lim inf_{𝑛 → ∞} 1 log𝑛

∑𝑛 𝑘=1

𝛼_𝑘

𝑘 ≥ 1 a.s.

(58)

Thus

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

𝛼_𝑘

𝑘 = 1 a.s. (59)

This completes the proof ofTheorem 2.

Proof ofTheorem 5. Let𝑘 < 𝑙 − 𝑙^𝛼,1 ≤ 𝑘 < 𝑙, and𝜀_𝑙 = 𝑙^3𝛼/2; we have

Cov(𝛼_𝑘, 𝛼_𝑙)

≤ 1

𝑝_𝑙𝑝_𝑘 (𝑃 (𝑎_𝑙− 𝑏_𝑘≤ 𝑆_𝑙− 𝑆_𝑘+𝑙^𝛼+ 𝑆_𝑘+𝑙^𝛼− 𝑆_𝑘

< 𝑏_𝑙− 𝑎_𝑘, 𝑎_𝑘≤ 𝑆_𝑘 < 𝑏_𝑘) − 𝑝_𝑙𝑝_𝑘)

≤ 1

𝑝_𝑙𝑝_𝑘 (𝑃 (𝑎_𝑙− 𝑏_𝑘≤ 𝑆_𝑙− 𝑆_𝑘+𝑙^𝛼+ 𝑆_𝑘+𝑙^𝛼− 𝑆_𝑘

< 𝑏_𝑙− 𝑎_𝑘) 𝑃 (𝑎_𝑘≤ 𝑆_𝑘< 𝑏_𝑘) − 𝑝_𝑙𝑝_𝑘)

≤ 1

𝑝_𝑙(𝑃 (𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑙< 𝑆_𝑙− 𝑆_𝑘+𝑙^𝛼 < 𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑙)

−𝑝_𝑙+ 𝑃 (󵄨󵄨󵄨󵄨𝑆^{𝑘+𝑙𝛼}− 𝑆_𝑘󵄨󵄨󵄨󵄨 ≥ 𝜀^𝑙))

≤ 1

𝑝_𝑙(𝑃 (𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑙< 𝑆_{𝑙−𝑘−𝑙}𝛼< 𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑙)

−𝑝_𝑙+ 𝑃 (󵄨󵄨󵄨󵄨𝑆^𝑙𝛼󵄨󵄨󵄨󵄨 ≥ 𝜀^𝑙)) .

(60)

ApplyingLemma 9, (33), and (35), we obtain 𝑃 (𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑙< 𝑆_{𝑙−𝑘−𝑙}^𝛼< 𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑙) − 𝑝_𝑙

≤ (Φ ( 𝑏_𝑙− 𝑎_𝑘+ 𝜀_𝑙

(𝑙 − 𝑘 − 𝑙^𝛼)^1/2) − Φ ( 𝑏_𝑙 𝑙^1/2)) + (Φ ( 𝑎_𝑙

𝑙^1/2) − Φ ( 𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑙 (𝑙 − 𝑘 − 𝑙^𝛼)^1/2)) + 𝑐 ( 1

𝑙^1/5+ 1

(𝑙 − 𝑘 − 𝑙^𝛼)^1/5) .

(61)

HenceLemma 11, (60), and (61) imply that

∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

Cov(𝛼_𝑘, 𝛼_𝑙)

𝑘𝑙 = 𝑂 (log𝑛) . (62)

On the other hand,

∑

1≤𝑘<𝑙≤𝑛 𝑘>𝑙−𝑙^𝛼

Cov(𝛼_𝑘, 𝛼_𝑙)

𝑘𝑙 = 𝑂 (log𝑛) , (63)

because𝑙 − 𝑙^𝛼 < 𝑘 < 𝑙, 𝑙^𝛾 − (𝑙 − 𝑙^𝛼)^𝛾 → 0as𝑙 → ∞for 𝛾 < 1, 𝛼 < 1.

But Var(𝛼_𝑘) = 0if𝑝_𝑘= 0and Var(𝛼_𝑘) = 1 − 𝑝_𝑘

𝑝_𝑘 ≤ 1

𝑝_𝑘 if𝑝_𝑘 ̸= 0. (64) Then

∑𝑛 𝑘=1

Var(𝛼_𝑘)

𝑘² ≤ ∑

1≤𝑘≤𝑛 𝑝𝑘 ̸= 0

1 𝑘²𝑝_𝑘 ≤∑^𝑛

𝑘=1

1

𝑘^5/2−𝛼 = 𝑂 (log𝑛) . (65) Noting that

Var(∑^𝑛

𝑘=1

𝛼_𝑘 𝑘)

=∑^𝑛

𝑘=1

Var(𝛼_𝑘) 𝑘² + 2 ∑

1≤𝑘<𝑙≤𝑛 𝑘>𝑙−𝑙^𝛼

Cov(𝛼_𝑘, 𝛼_𝑙)

𝑘𝑙 + 2 ∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

Cov(𝛼_𝑘, 𝛼_𝑙)

𝑘𝑙 ,

(66)

thus (62)–(66) imply that Var(∑^𝑛

𝑘=1

𝛼_𝑘

𝑘) = 𝑂 (log𝑛) , as𝑛 󳨀→ ∞. (67) Hence applyingRemark 7, we have

𝑛 → ∞lim 1 log𝑛

∑𝑛 𝑘=1

𝛼_𝑘

𝑘 = 1 a.s. (68)

This completes the proof ofTheorem 5.

Appendix

Proof ofLemma 6. Let𝜇_𝑛= ∑^𝑛_𝑘=1𝑑_𝑘𝜉_𝑘, and then∀𝜀 > 0 𝑃 (󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝜇_𝑛

𝐷_𝑛 −𝐸𝜇_𝑛

𝐷_𝑛 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≥ 𝜀) ≤ Var(𝜇_𝑛/𝐷_𝑛)

𝜀² ≤ 𝑐(log𝐷_𝑛)^−𝛽. (A.1) Let𝑟 < 1,𝑟𝛽 > 1, and𝑛_𝑘 = inf{𝑛_𝑘, 𝐷_𝑛𝑘 ≥ exp(𝑘^𝑟)}, and then𝐷_𝑛𝑘 ≥exp(𝑘^𝑟), 𝐷_𝑛𝑘−1<exp(𝑘^𝑟), for𝐷_𝑛 ∼ 𝐷_𝑛−1; we get

1 ≤ 𝐷_𝑛𝑘

exp(𝑘^𝑟) ∼ 𝐷_𝑛𝑘−1

exp(𝑘^𝑟) < 1 󳨀→ 1, (A.2) that is,

𝐷_𝑛𝑘 ∼exp(𝑘^𝑟) , (A.3) and thus

𝐷_𝑛𝑘

𝐷_𝑛𝑘−1 ∼ exp(𝑘^𝑟) exp((𝑘 − 1)^𝑟)

=exp(𝑘^𝑟[1 − (1 −1

𝑘)^𝑟]) ∼exp(𝑘^𝑟⋅ 𝑟 ⋅1 𝑘)

=exp(𝑟 ⋅ 𝑘^𝑟−1) .

(A.4) On account of𝑟 < 1, then

𝐷_𝑛𝑘

𝐷_𝑛𝑘−1 ∼exp(𝑟 ⋅ 𝑘^𝑟−1) 󳨀→ 1, as𝑘 󳨀→ ∞,

∑∞ 𝑘=1

𝑃 (󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝜇_𝑛𝑘 𝐷_𝑛𝑘 −𝐸𝜇_𝑛𝑘

𝐷_𝑛𝑘 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨≥ 𝜀)

≤ 𝑐∑^∞

𝑘=1

1

(log𝐷_𝑛)^−𝛽 ≤ 𝑐∑^∞

𝑘=1

1 𝑘^𝑟𝛽 < ∞.

(A.5)

By the Borel-Cantelli lemma, 𝜇_𝑛𝑘 𝐷_𝑛𝑘 −𝐸𝜇_𝑛𝑘

𝐷_𝑛𝑘 󳨀→ 0 a.s. (A.6)

Since

𝐸𝜇_𝑛𝑘

𝐷_𝑛𝑘 = ∑^𝑛_𝑘=1^𝑘 𝑑_𝑘

𝐷_𝑛𝑘 = 1, (A.7)

thus

𝜇_𝑛𝑘

𝐷_𝑛𝑘 󳨀→ 1, a.s. (A.8)

Now for𝑛_𝑘−1≤ 𝑛 < 𝑛_𝑘, for𝐷_𝑛 ↑ ∞, 𝐷_𝑛𝑘/𝐷_𝑛𝑘−1 → 1, and by 𝜉_𝑘≥ 0, then𝜇_𝑛↑, and we have

1 ←󳨀 𝐷_𝑛𝑘−1 𝐷_𝑛𝑘

𝜇_𝑛𝑘−1 𝐷_𝑛𝑘−1 ≤ 𝜇_𝑛

𝐷_𝑛 ≤ 𝜇_𝑛𝑘 𝐷_𝑛𝑘

𝐷_𝑛𝑘

𝐷_𝑛𝑘−1 󳨀→ 1 a.s. (A.9) hence

𝜇_𝑛

𝐷_𝑛 󳨀→ 1 a.s. (A.10)

This completes the proof ofLemma 6.

Proof ofLemma 10. ApplyingLemma 9, (33), and noting the conditions of (19) and0 < 𝛼 < 1/7, we get

𝑝_𝑘= 𝑃 (𝑎_𝑘≤ 𝑆_𝑘 ≤ 𝑏_𝑘)

≤ (Φ ( 𝑏_𝑘

𝑘^1/2) − Φ ( 𝑎_𝑘

𝑘^1/2)) + 𝑐 1 𝑘^1/5

≤ 𝑐 (𝑘^1/2−𝛼 𝑘^1/2 + 1

𝑘^1/5) ≤ 𝑐₂^󸀠𝑘^−𝛼.

(A.11)

ApplyingLemma 9, (34), and noting the conditions of (19) and0 < 𝛼 < 1/7, we have

𝑝_𝑘= 𝑃 (𝑎_𝑘≤ 𝑆_𝑘 ≤ 𝑏_𝑘)

≥ (Φ ( 𝑏_𝑘

𝑘^1/2) − Φ ( 𝑎_𝑘

𝑘^1/2)) − 𝑐 1 𝑘^1/5

≥ 𝑐 (𝑘^1/2−𝛼 𝑘^1/2 + 1

𝑘^1/5) ≥ 𝑐₁^󸀠𝑘^−𝛼.

(A.12)

ThusLemma 10immediately follows from (A.11) and (A.12).

Proof ofLemma 11.By Lemma 10, Chebyshev’s inequality, and noting the condition of0 < 𝛼 < 1/7, we have

∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1

𝑘𝑙𝑝_𝑙𝑃 (󵄨󵄨󵄨󵄨𝑆^𝑙𝛼󵄨󵄨󵄨󵄨 ≥ 𝜀^𝑙)

≤ ∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1 𝑘𝑙𝑝_𝑙

Var(𝑆_𝑙^𝛼) 𝑙^3𝛼 ≤ 𝑐∑^𝑛

𝑙=1

1 𝑙^1+𝛼

𝑙−𝑙^𝛼

∑

𝑘=1

1 𝑘

≤ 𝑐∑^𝑛

𝑙=1

log(𝑙 − 𝑙^𝛼)

𝑙^1+𝛼 = 𝑂 (log𝑛) .

(A.13)

It proves (29). ByLemma 10and0 < 𝛼 < 1/7, we get

∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1 𝑘𝑙𝑝_𝑙

1 (𝑙 − 𝑘 − 𝑙^𝛼)^1/5

≤ 𝑐∑^𝑛

𝑙=1

1

𝑙^1−𝛼( ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1 𝑘(𝑙 − 𝑙^𝛼− 𝑘)^1/5

+ ∑

(𝑙−𝑙^𝛼)/2<𝑘<𝑙−𝑙^𝛼

1

𝑘(𝑙 − 𝑙^𝛼− 𝑘)^1/5)

≤ 𝑐∑^𝑛

𝑙=1

1

𝑙^1−𝛼( 1

(𝑙 − 𝑙^𝛼)^1/5 ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1 𝑘 + 1

𝑙 − 𝑙^𝛼 ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1 𝑘^1/5)

≤ 𝑐∑^𝑛

𝑙=1

log𝑙 𝑙^1−𝛼+1/5 ≤ 𝑐∑^𝑛

𝑙=1

1

𝑙 = 𝑂 (log𝑛) .

(A.14)

It proves (30). ApplyingLemma 9, (33), (35),𝜀_𝑙 = 𝑙^3𝛼/2, and noting the condition of0 < 𝛼 < 1/7, we obtain

∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1

𝑘𝑙𝑝_𝑙󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨Φ ( 𝑎_𝑙− 𝑏_𝑘− 𝜀_𝑙

(𝑙 − 𝑘 − 𝑙^𝛼)^1/2) − Φ ( 𝑎_𝑙 𝑙^1/2)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤ 𝑐 ∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

−𝑎_𝑙

𝑘𝑙𝑝_𝑙( 1

(𝑙 − 𝑘 − 𝑙^𝛼)^1/2− 1

√𝑙)

+ 𝑐 ∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1 𝑘𝑙𝑝_𝑙

𝑏_𝑘 (𝑙 − 𝑘 − 𝑙^𝛼)^1/2 + 𝑐 ∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1 𝑘𝑙𝑝_𝑙

𝜀_𝑙

(𝑙 − 𝑘 − 𝑙^𝛼)^1/2 := Σ₁+ Σ₂+ Σ₃. (A.15)

Now applying the same procedure as before, we have

Σ₁≤ 𝑐 ∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

−𝑎_𝑙

𝑘𝑙𝑝_𝑙( 1

𝑙^1−𝛼(𝑙 − 𝑘 − 𝑙^𝛼)^1/2 + 𝑘 𝑙(𝑙 − 𝑘 − 𝑙^𝛼)^1/2)

≤ 𝑐 ∑

1≤𝑙≤𝑛

1 𝑙^3/2−𝛼

× ( 1

(l− 𝑙^𝛼)^1/2 ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1 𝑘+ 1

𝑙 − 𝑙^𝛼 ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1 𝑘^1/2) + 𝑐 ∑

1≤𝑙≤𝑛

1 𝑙^3/2 ∑

1≤𝑘<𝑙−𝑙^𝛼

1 (𝑙 − 𝑘 − 𝑙^𝛼)^1/2

≤ 𝑐 ∑

1≤𝑙≤𝑛

log𝑙 𝑙^2−𝛼 ≤ 𝑐 ∑

1≤𝑙≤𝑛

1

𝑙 = 𝑂 (log𝑛) , Σ₂≤ 𝑐 ∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1

𝑙^1−𝛼𝑘^1/2+𝛼(𝑙 − 𝑙^𝛼− 𝑘)^1/2

≤ 𝑐 ∑

1≤𝑙≤𝑛

1

𝑙^1−𝛼( 1

(𝑙 − 𝑙^𝛼)^1/2 ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1 𝑘^1/2+𝛼

+ 1

(𝑙 − 𝑙^𝛼)^1/2+𝛼 ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1 𝑘^1/2)

≤ 𝑐 ∑

1≤𝑙≤𝑛

1

𝑙^1−𝛼( 1

(𝑙 − 𝑙^𝛼)^1/2𝑙^1/2−𝛼+ 1

(𝑙 − 𝑙^𝛼)^1/2+𝛼(𝑙 − 𝑙^𝛼)^1/2)

≤ 𝑐 ∑

1≤𝑙≤𝑛

1

𝑙 = 𝑂 (log𝑛) .

(A.16)

Noting that0 < 𝛼 < 1/7, we deduce Σ₃≤ 𝑐 ∑

1≤𝑘<𝑙≤𝑛 𝑘<𝑙−𝑙^𝛼

1

𝑙^{1−(7/2)𝛼}𝑘^1/2(𝑙 − 𝑙^𝛼− 𝑘)^1/2

≤ 𝑐 ∑

1≤𝑙≤𝑛

1 𝑙^{1−(7/2)𝛼}

× ( 1

(𝑙 − 𝑙^𝛼)^1/2 ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1

𝑘+ 1

(𝑙 − 𝑙^𝛼) ∑

1≤𝑘<(𝑙−𝑙^𝛼)/2

1 𝑘^1/2)

≤ 𝑐 ∑

1≤𝑙≤𝑛

log𝑙

𝑙^{3/2−(7/2)𝛼} ≤ 𝑐 ∑

1≤𝑙≤𝑛

1

𝑙 = 𝑂 (log𝑛) .

(A.17) It proves (31). The proof of (32) is similar to the proof of (31).

This completes the proof ofLemma 11.

Acknowledgments

The authors are very grateful to the academic editor, pro- fessor Ying Hu, and the two anonymous reviewers for their valuable comments and helpful suggestions, which significantly contributed to improving the quality of this paper. This work is jointly supported by National Natural Science Foundation of China (11061012,71271210), Project Supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ((2011)47), the Guangxi China Science Foundation (2013GXNSFDA019001).

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