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

Abstract

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 Csáki 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≥2 are said to be negatively associated (NA) if for every pair of disjoint subsets 𝐴₁ and 𝐴₂ of {1,2,…,𝑛}, Cov𝑓₁𝑋_𝑖,𝑖∈𝐴₁,𝑓₂𝑋_𝑗,𝑗∈𝐴₂≤0, (1) where 𝑓₁ and 𝑓₂ are increasing for every variable (or decreasing 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 applications 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 with 𝐸𝑋₁=0,0<𝐸𝑋²₁<∞. Define 𝑆_𝑛=∑^𝑛_𝑗=1𝑋_𝑗, 𝜎²_𝑛∶=Var𝑆_𝑛,𝜎²∶=Var𝑋₁+2^∞_𝑗=2Cov𝑋₁,𝑋_𝑗. (2)(3)(1)Newman [4] and Matuła [5] showed that NA stationary sequences satisfy the central limit theorem (CLT) under 𝜎²>0, that is, sup_𝑥∈𝑅𝑃𝑆_𝑛𝜎_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, lim_𝑛→∞1log𝑛^𝑛_𝑘=11𝑘𝐼𝑆_𝑘≤𝑥𝜎𝑘^1/2=Φ(𝑥)a.s.∀𝑥∈ℝ, (5) where Φ(𝑥) is the standard normal distribution function and 𝐼{𝐴} denotes the indicator of the event 𝐴.

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 as 𝛼,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 Csáki 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 with 𝐸|𝑋₁|³<∞, let 𝐸𝑋₁=0. 𝑎_𝑘,𝑏_𝑘 satisfy −∞≤𝑎_𝑘≤0≤𝑏_𝑘≤∞,𝑘=1,2,…, (6) and assume that ^𝑛_𝑘=1log𝑘𝑘^3/2𝑃𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘=𝑂log𝑛,𝑎𝑠𝑛⟶∞, (7) and then lim_𝑛→∞1log𝑛^𝑛_𝑘=1𝐼𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘𝑘𝑃𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘=1a.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 of 𝜌-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 Csáki [15] and Földes [16].

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

In the next section we present the exact results, postponing some technical lemmas and the proofs to Section 3.

2. Main Results

Assume in the following that {𝑋_𝑛,𝑛≥1} is a strictly stationary sequence of NA random variables with 𝐸𝑋₁=0,0<𝐸𝑋²₁<∞. We consider the limit behavior of the logarithmic average lim_𝑛→∞1log𝑛^𝑛_𝑘=1𝐼𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘𝑘𝑃𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘 (9) with −∞≤𝑎_𝑘≤0≤𝑏_𝑘≤∞, where the terms in the sum above are defined to be 1 if 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∶=𝑃𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘,𝛼_𝑘𝐼𝑎∶=^𝑘≤𝑆_𝑘<𝑏_𝑘𝑝_𝑘,if𝑝_𝑘≠0,1,if𝑝_𝑘=0. (10) So we need to investigate the limit behavior of 𝜇_𝑛∶=^𝑛_𝑘=1𝛼_𝑘𝑘 (11) under certain conditions.

In our considerations, we will need the following Cox-Grimmett coefficient which describes the covariance structure of the sequence 𝑢(𝑛)∶=sup_{𝑘∈𝑁𝑗∶|𝑗−𝑘|≥𝑛}Cov𝑋_𝑗,𝑋_𝑘,𝑛∈𝑁∪{0}. (12) We remark that for a stationary sequence of NA random variables 𝑢(𝑛)=−2^∞_𝑘=𝑛+1Cov𝑋₁,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_𝑛 denotes 𝜉_𝑛/𝜂_𝑛→1,𝑛→∞. 𝜉_𝑛=𝑂(𝜂_𝑛) denotes that there exists a constant 𝑐>0 such that 𝜉_𝑛≤𝑐𝜂_𝑛 for sufficiently large 𝑛. The symbols 𝑐,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 with 𝐸𝑋₁=0,𝐸|𝑋₁|³<∞ and let 𝜎²>0.𝑎_𝑘,𝑏_𝑘 satisfy (6). Assume that ^∞_𝑛=1𝑢(𝑛)<∞, (14) and for some 𝛽>1, _{𝑝1≤𝑘≤𝑛𝑘≠0}log𝑘^1/3𝑘𝑝_𝑘=𝑂log𝑛²loglog𝑛^−𝛽. (15) Then we have lim_𝑛→∞𝜇_𝑛log𝑛=1,a.s., (16) where 𝜇_𝑛 is defined by (11).

Remark 3. Let 𝑎_𝑘=−∞ and 𝑏_𝑘=𝑥𝜎𝑘^1/2 in (6). By the central limit theorem (4), we have 𝑝_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=0or𝑝_𝑘≥𝑐loglog𝑘^𝛽log𝑘^2/3 (17) holds, then the condition (15) is satisfied. In fact, letting 0<𝛿<1, we have _{𝑝1≤𝑘≤𝑛𝑘≠0}log𝑘^1/3𝑘𝑝_𝑘≤𝑐_{1≤𝑘≤𝑛}loglog𝑘^−𝛽log𝑘^𝛿log𝑘^1−𝛿𝑘≤𝑐loglog𝑛^−𝛽log𝑛^𝛿_{1≤𝑘≤𝑛}log𝑘^1−𝛿𝑘≤𝑐loglog𝑛^−𝛽log𝑛²=𝑂log𝑛²loglog𝑛^−𝛽. (18)

In the given theorem below, we strengthen the condition (6) on 𝑎_𝑘 and 𝑏_𝑘. Meanwhile, as a compensation, we do not need to impose restricting condition (15) on 𝑝_𝑘.

Theorem 5. Let {𝑋_𝑛,𝑛≥1} be a strictly stationary sequence of NA random variables with 𝐸𝑋₁=0,𝐸|𝑋₁|³<∞, and 𝜎²>0, and let 𝑎_𝑘 and 𝑏_𝑘 satisfy −𝑐₁𝑘^1/2−𝛼≤𝑎_𝑘≤−𝑐₂𝑘^1/2−𝛼,𝑐₃𝑘^1/2−𝛼≤𝑏_𝑘≤𝑐₄𝑘^1/2−𝛼, (19) where 0<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 exists 𝑑_𝑘≥0 such that 𝐷_𝑛=∑^𝑛_𝑘=1𝑑_𝑘↑∞,𝐷_𝑛/𝐷_𝑛−1→1, and Var^𝑛_𝑘=1𝑑_𝑘𝜉_𝑘≤𝑐𝐷²_𝑛log𝐷_𝑛^−𝛽, (21) with some 𝛽>1 and positive constant 𝑐, and then lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝜉_𝑘=1a.s. (22)

Remark 7. Let 𝑑_𝑘=1/𝑘 in (21), and then 𝐷_𝑛=∑^𝑛_𝑘=11/𝑘∼log𝑛. Thus, if Var^𝑛_𝑘=11𝑘𝜉_𝑘≤𝑐log𝑛²loglog𝑛^−𝛽, (23) with some 𝛽>1 and positive constant 𝑐, then lim_𝑛→∞1log𝑛^𝑛_𝑘=11𝑘𝜉_𝑘=1a.s. (24)

The following Lemma 8 is 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 any 𝜀>0, there exists a 𝑘₀=𝑘₀(𝜀,𝜔) for which (1−𝜀)𝜉_𝑘≤𝜂_𝑘≤(1+𝜀)𝜉_𝑘,𝑘>𝑘₀. (25) Then we have also lim_𝑛→∞1log𝑛^𝑛_𝑘=11𝑘𝜂_𝑘=1a.s. (26)

The following Lemma 9 is an easy corollary to the Corollary 2.2 in Matuła [5] under strictly stationary condition, which studies the rate of convergence in the CLT under negative 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 with 𝐸𝑋₁=0,𝐸|𝑋₁|³<∞,𝜎²>0 satisfying (14). Then one has sup_𝑥∈𝑅𝑃𝑆_𝑛𝜎_𝑛<𝑥−Φ(𝑥)=𝑂𝑛^−1/5. (27)

Lemma 10. If the conditions of Lemma 9 hold, 𝑎_𝑘 and 𝑏_𝑘 satisfy (19). Then one has 𝑐^′₁𝑘^−𝛼≤𝑝_𝑘≤𝑐^′₂𝑘^−𝛼𝑘≥𝑘₀, (28) where 𝛼 is as (19).

Lemma 11. If the conditions of Lemma 9 hold, 𝑎_𝑘 and 𝑏_𝑘 satisfy (19), and 𝛼 is as (19). Assume that 𝜀_𝑙=𝑙^3𝛼/2, and then the following asymptotic relations hold: _{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑘𝑙𝑝_𝑙𝑃𝑆_𝑙^𝛼≥𝜀_𝑙=𝑂log𝑛_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼},1𝑘𝑙𝑝_𝑙1𝑙−𝑘−𝑙^𝛼1/5=𝑂log𝑛_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼},1𝑘𝑙𝑝_𝑙Φ𝑎_𝑙−𝑏_𝑘−𝜀_𝑙𝑙−𝑘−𝑙^𝛼1/2𝑎−Φ^𝑙𝑙^1/2=𝑂log𝑛_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼},1𝑘𝑙𝑝_𝑙Φ𝑏_𝑙−𝑎_𝑘+𝜀_𝑙𝑙−𝑘−𝑙^𝛼1/2𝑏−Φ^𝑙𝑙^1/2=𝑂log𝑛. (29)(30)(31)(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)−Φ𝑦≤𝑐𝑥−𝑦,forevery𝑥,𝑦∈ℝ (33) with some constant 𝑐. Moreover, for each 𝑘>0, there exists 𝑐₁=𝑐₁(𝑘), such that Φ(𝑥)−Φ𝑦≥𝑐₁𝑥−𝑦,forevery𝑥,𝑦∈ℝ,|𝑥|+𝑦≤𝑘. (34)

Let {𝑋_𝑛,𝑛≥1} be a strictly stationary sequence of NA random variables with 𝜎²>0; we can immediately get 𝜎²_𝑛∼𝑛𝜎², that is, 𝑐₁𝑛≤Var𝑆_𝑛=𝜎²_𝑛≤𝑐₂𝑛 (35) for some constant 𝑐₁,𝑐₂>0 and sufficiently large 𝑛.

Proof of Theorem 2. First assume that 𝑏_𝑘−𝑎_𝑘≤𝑐𝑘^1/2,𝑘=1,2,… (36) with some constant 𝑐. Let 1≤𝑘<𝑙 and 𝜀_𝑘=𝑘^1/2(log𝑘)^1/3.
If either 𝑝_𝑘=0 or 𝑝_𝑙=0, then obviously Cov(𝛼_𝑘,𝛼_𝑙)=0, and so we may assume that 𝑝_𝑘𝑝_𝑙≠0; then, we have Cov𝛼_𝑘,𝛼_𝑙=1𝑝_𝑙𝑝_𝑘𝑃𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘,𝑎_𝑙≤𝑆_𝑙<𝑏_𝑙−𝑝_𝑙𝑝_𝑘≤1𝑝_𝑙𝑝_𝑘𝑃𝑎_𝑙−𝑏_𝑘≤𝑆_𝑙−𝑆_𝑘<𝑏_𝑙−𝑎_𝑘,𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘−𝑝_𝑙𝑝_𝑘≤1𝑝_𝑙𝑝_𝑘𝑃𝑎_𝑙−𝑏_𝑘≤𝑆_𝑙−𝑆_𝑘<𝑏_𝑙−𝑎_𝑘×𝑃𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘−𝑝_𝑙𝑝_𝑘=1𝑝_𝑙𝑃𝑎_𝑙−𝑏_𝑘≤𝑆_𝑙−𝑆_𝑘<𝑏_𝑙−𝑎_𝑘−𝑝_𝑙≤1𝑝_𝑙𝑃𝑎_𝑙−𝑏_𝑘−𝜀_𝑘<𝑆_𝑙<𝑏_𝑙−𝑎_𝑘+𝜀_𝑘−𝑝_𝑙+𝑃𝑆_𝑘≥𝜀_𝑘≤1𝑝_𝑙𝑃𝑎_𝑙−𝑏_𝑘−𝜀_𝑘≤𝑆_𝑙<𝑎_l+𝑃𝑏_𝑙≤𝑆_𝑙<𝑏_𝑙−𝑎_𝑘+𝜀_𝑘+𝑃𝑆_𝑘≥𝜀_𝑘. (37)
Applying Lemma 9, (33), (35), and (36) and noting that 𝜀_𝑘=𝑘^1/2(log𝑘)^1/3, we obtain 𝑃𝑎_𝑙−𝑏_𝑘−𝜀_𝑘≤𝑆_𝑙<𝑎_𝑙+𝑃𝑏_𝑙≤𝑆_𝑙<𝑏_𝑙−𝑎_𝑘+𝜀_𝑘≤Φ𝑎_𝑙𝜎_𝑙𝑎−Φ^𝑙−𝑏_𝑘−𝜀_𝑘𝜎_𝑙+Φ𝑏_𝑙−𝑎_𝑘+𝜀_𝑘𝜎_𝑙𝑏−Φ^𝑙𝜎_𝑙1+𝑐𝑙^1/5≤𝑏_𝑘+𝜀_𝑘𝜎_𝑙+−𝑎_𝑘+𝜀_𝑘𝜎_𝑙1+𝑐𝑙^1/5=𝑏_𝑘−𝑎_𝑘𝜎_𝑙+2𝜀_𝑘𝜎_𝑙1+𝑐𝑙^1/5𝑘≤𝑐^1/2𝑙^1/2+𝜀_𝑘𝑙^1/2+1𝑙^1/5𝑘≤𝑐^1/2log𝑘^1/3𝑙^1/2+1𝑙^1/5. (38) By the condition of (15), we have ^𝑛𝑙_𝑙=1𝑘=11𝑘𝑙𝑝_𝑙𝑘^1/2log𝑘^1/3𝑙^1/2≤^𝑛_𝑙=11𝑙^3/2𝑝_𝑙^𝑙_𝑘=1log𝑘^1/3𝑘^1/2≤𝑐^𝑛_𝑙=1log𝑙^1/3𝑙^3/2𝑝_𝑙^𝑙_𝑘=11𝑘^1/2≤𝑐^𝑛_𝑙=1log𝑙^1/3𝑙^3/2𝑝_𝑙𝑙^1/2≤𝑐^𝑛_𝑙=1log𝑙^1/3𝑙𝑝_𝑙=𝑂log𝑛²loglog𝑛^−𝛽,_{𝑛𝑙=1𝑙−1𝑘=1}1𝑘𝑙𝑝_𝑙1𝑙^1/5=^𝑛_𝑙=11𝑙^6/5𝑝_𝑙^𝑙−1_𝑘=11𝑘≤𝑐^𝑛_𝑙=11𝑙^6/5𝑝_𝑙log𝑙≤𝑐^𝑛_𝑙=1log𝑙^1/3𝑙𝑝_𝑙=𝑂log𝑛²loglog𝑛^−𝛽. (39)
By Chebyshev’s inequality and the condition of (15) and (35), we obtain ^𝑛𝑙_𝑙=1𝑘=11𝑘𝑙𝑝_𝑙𝑃𝑆_𝑘≥𝜀_𝑘≤^𝑛_𝑙=11𝑙𝑝_𝑙^𝑙_𝑘=1Var𝑆_𝑘𝑘𝜀²_𝑘≤𝑐^𝑛_𝑙=11𝑙𝑝_𝑙^𝑙_𝑘=11𝑘log𝑘^2/3≤𝑐^𝑛_𝑙=1log𝑙^1/3𝑙𝑝_𝑙=𝑂log𝑛²loglog𝑛^−𝛽. (40) Hence (37)–(40) imply thatCov𝛼^𝑛_𝑙=1^𝑙−1_𝑘=1𝑘,𝛼_𝑙𝑘𝑙=𝑂log𝑛²loglog𝑛^−𝛽. (41)
But Var(𝛼_𝑘)=0 if 𝑝_𝑘=0 and Var𝛼_𝑘=1−𝑝_𝑘𝑝_𝑘≤1𝑝_𝑘if𝑝_𝑘≠0. (42)
Thus Var𝛼^𝑛_𝑘=1𝑘𝑘²≤_{𝑝1≤𝑘≤𝑛𝑘≠0}1𝑘²𝑝_𝑘≤_{𝑝1≤𝑘≤𝑛𝑘≠0}log𝑘^1/3𝑘𝑝_𝑘=𝑂log𝑛²loglog𝑛^−𝛽. (43) Equations (41) and (43) together imply that Var^𝑛_𝑘=1𝛼_𝑘𝑘=𝑂log𝑛²loglog𝑛^−𝛽,as𝑛→∞. (44)
Hence applying Remark 7, our theorem is proved under the restricting condition (36).
Now we drop the restricting condition (36). Fix 𝑥>0 and define 𝑎_𝑘=max𝑎_𝑘,−𝑥𝜎𝑘^1/2,𝑏̃_𝑘=min𝑏_𝑘,𝑥𝜎𝑘^1/2,𝑝_𝑘=𝑃𝑎_𝑘≤𝑆_𝑘<𝑏̃_𝑘, (45) where 𝜎 is defined by (3).
Clearly 𝑝_𝑘≤𝑝_𝑘, and so assuming 𝑝_𝑘≠0; then, also we have 𝑝_𝑘≠0, and thus 𝛼_𝑘=1𝑝_𝑘𝐼𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘=1𝑝_𝑘𝐼𝑎_𝑘≤𝑆_𝑘<𝑏̃_𝑘+𝐼𝑎_𝑘≤𝑆_𝑘<𝑎_𝑘+𝐼𝑏̃_𝑘≤𝑆_𝑘<𝑏_𝑘≤1𝑝_𝑘𝐼𝑎_𝑘≤𝑆_𝑘<𝑏̃_𝑘+1𝑝_𝑘𝐼𝑎_𝑘≤𝑆_𝑘<𝑎_𝑘+𝐼𝑏̃_𝑘≤𝑆_𝑘<𝑏_𝑘≤1𝑝_𝑘𝐼𝑎_𝑘≤𝑆_𝑘<𝑏̃_𝑘+𝐼𝑆_𝑘<−𝑥𝜎𝑘^1/2𝑃−𝑥𝜎𝑘^1/2≤𝑆_𝑘+𝐼𝑆<0^𝑘≥𝑥𝜎𝑘^1/2𝑃0≤𝑆_𝑘<𝑥𝜎𝑘^1/2. (46)
By (35) and the central limit theorem for NA random variables (4), that is, sup_𝑥∈𝑅𝑃𝑆_𝑛𝜎_𝑛<𝑥−Φ(𝑥)=𝑜(1), (47)
we obtain lim_𝑘→∞𝑃−𝑥𝜎𝑘^1/2≤𝑆_𝑘<0=lim_𝑘→∞𝑃−𝑥𝜎_𝑘≤𝑆_𝑘<0=Φ(0)−Φ(−𝑥),lim_𝑘→∞𝑃0≤𝑆_𝑘<𝑥𝜎𝑘^1/2=lim_𝑘→∞𝑃0≤𝑆_𝑘<𝑥𝜎_𝑘=Φ(𝑥)−Φ(0). (48)
Applying the almost sure central limit theorem for NA random variables (5), that is, lim_𝑛→∞1log𝑛^𝑛_𝑘=11𝑘𝐼𝑆_𝑘≤𝑥𝜎𝑘^1/2=Φ(𝑥)a.s.∀𝑥∈ℝ, (49) Lemma 8, and (48), we have lim_𝑛→∞1log𝑛^𝑛_𝑘=1𝐼𝑆_𝑘<−𝑥𝜎𝑘^1/2𝑘𝑃−𝑥𝜎𝑘^1/2≤𝑆_𝑘=Φ()<0−𝑥Φ(0)−Φ(−𝑥)a.s.,lim_𝑛→∞1log𝑛^𝑛_𝑘=1𝐼𝑆_𝑘>𝑥𝜎𝑘^1/2𝑘𝑃0≤𝑆_𝑘<𝑥𝜎𝑘^1/2=1−Φ(𝑥)Φ(𝑥)−Φ(0)a.s. (50) Since 𝑎_𝑘 and 𝑏̃_𝑘 satisfy (36), we get lim_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘=1a.s., (51) where 𝛼_𝑘=𝐼𝑎_𝑘≤𝑆_𝑘<𝑏̃_𝑘𝑝_𝑘,if𝑝_𝑘𝑝≠0,1,if_𝑘=0. (52) Equations (46) and (50)–(51) together imply thatlimsup_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘(𝑥)≤1+21−ΦΦ(𝑥)−Φ(0)a.s. (53) On the other hand, if 𝑝_𝑘≠0, then we have 1𝑝_𝑘𝐼𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘≥1𝑝_𝑘𝐼𝑎_𝑘≤𝑆_𝑘<𝑏̃_𝑘𝑝1−^𝑘−𝑝_𝑘𝑝_𝑘≥𝛼_𝑘𝑃𝑆1−^𝑘<−𝜎𝑥𝑘^1/2+𝑃𝑆_𝑘>𝜎𝑥𝑘^1/2min𝑃0≤𝑆_𝑘<𝜎𝑥𝑘^1/2,𝑃−𝜎𝑥𝑘^1/2≤𝑆_𝑘,<0 (54) and by the central limit theorem, lim_𝑘→∞𝑃𝑆_𝑘<−𝜎𝑥𝑘^1/2+𝑃𝑆_𝑘>𝜎𝑥𝑘^1/2min𝑃0≤𝑆_𝑘<𝜎𝑥𝑘^1/2,𝑃−𝜎𝑥𝑘^1/2≤𝑆_𝑘(𝑥)<0=1−21−ΦΦ(𝑥)(0).−Φ (55) Applying Lemma 8, (51) and (54) imply that liminf_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘(𝑥)≥1−21−ΦΦ(𝑥)−Φ(0)a.s., (56) and hence (𝑥)1+21−ΦΦ(𝑥)−Φ(0)≥limsup_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘≥liminf_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘(𝑥)≥1−21−ΦΦ(𝑥)−Φ(0)a.s. (57) By the arbitrariness of 𝑥, let 𝑥→∞ in (57); we have 1≥limsup_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘≥liminf_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘≥1a.s. (58) Thus lim_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘=1a.s. (59) This completes the proof of Theorem 2.

Proof of Theorem 5. Let 𝑘<𝑙−𝑙^𝛼, 1≤𝑘<𝑙, and 𝜀_𝑙=𝑙^3𝛼/2; we have Cov𝛼_𝑘,𝛼_𝑙≤1𝑝_𝑙𝑝_𝑘𝑃𝑎_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≤1𝑝_𝑙𝑝_𝑘𝑃𝑎_𝑙−𝑏_𝑘≤𝑆_𝑙−𝑆_𝑘+𝑙^𝛼+𝑆_𝑘+𝑙^𝛼−𝑆_𝑘<𝑏_𝑙−𝑎_𝑘𝑃𝑎_𝑘≤𝑆_𝑘<𝑏_𝑘−𝑝_𝑙𝑝_𝑘≤1𝑝_𝑙𝑃𝑎_𝑙−𝑏_𝑘−𝜀_𝑙<𝑆_𝑙−𝑆_𝑘+𝑙^𝛼<𝑏_𝑙−𝑎_𝑘+𝜀_𝑙−𝑝_𝑙+𝑃𝑆_𝑘+𝑙^𝛼−𝑆_𝑘≥𝜀_𝑙≤1𝑝_𝑙𝑃𝑎_𝑙−𝑏_𝑘−𝜀_𝑙<𝑆_{𝑙−𝑘−𝑙}^𝛼<𝑏_𝑙−𝑎_𝑘+𝜀_𝑙−𝑝_𝑙+𝑃𝑆_𝑙^𝛼≥𝜀_𝑙. (60)
Applying Lemma 9, (33), and (35), we obtain 𝑃𝑎_𝑙−𝑏_𝑘−𝜀_𝑙<𝑆_{𝑙−𝑘−𝑙}^𝛼<𝑏_𝑙−𝑎_𝑘+𝜀_𝑙−𝑝_𝑙≤Φ𝑏_𝑙−𝑎_𝑘+𝜀_𝑙𝑙−𝑘−𝑙^𝛼1/2𝑏−Φ^𝑙𝑙^1/2+Φ𝑎_𝑙𝑙^1/2𝑎−Φ^𝑙−𝑏_𝑘−𝜀_𝑙𝑙−𝑘−𝑙^𝛼1/21+𝑐𝑙^1/5+1𝑙−𝑘−𝑙^𝛼1/5. (61) Hence Lemma 11, (60), and (61) imply that Cov𝛼_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼𝑘},𝛼_𝑙.𝑘𝑙=𝑂log𝑛 (62) On the other hand, Cov𝛼_{1≤𝑘<𝑙≤𝑛𝑘>𝑙−𝑙𝛼𝑘},𝛼_𝑙,𝑘𝑙=𝑂log𝑛 (63) because 𝑙−𝑙^𝛼<𝑘<𝑙,𝑙^𝛾−(𝑙−𝑙^𝛼)^𝛾→0 as 𝑙→∞ for 𝛾<1,𝛼<1.
But Var(𝛼_𝑘)=0 if 𝑝_𝑘=0 and Var𝛼_𝑘=1−𝑝_𝑘𝑝_𝑘≤1𝑝_𝑘if𝑝_𝑘≠0. (64) Then Var𝛼^𝑛_𝑘=1𝑘𝑘²≤_{𝑝1≤𝑘≤𝑛𝑘≠0}1𝑘²𝑝_𝑘≤^𝑛_𝑘=11𝑘^5/2−𝛼=𝑂log𝑛. (65) Noting that Var^𝑛_𝑘=1𝛼_𝑘𝑘=^𝑛_𝑘=1Var𝛼_𝑘𝑘²+2_{1≤𝑘<𝑙≤𝑛𝑘>𝑙−𝑙𝛼}Cov𝛼_𝑘,𝛼_𝑙𝑘𝑙+2_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}Cov𝛼_𝑘,𝛼_𝑙,𝑘𝑙 (66) thus (62)–(66) imply that Var^𝑛_𝑘=1𝛼_𝑘𝑘=𝑂log𝑛,as𝑛⟶∞. (67) Hence applying Remark 7, we havelim_𝑛→∞1log𝑛^𝑛_𝑘=1𝛼_𝑘𝑘=1a.s. (68) This completes the proof of Theorem 5.

Appendix

Proof of Lemma 6. Let 𝜇_𝑛=∑^𝑛_𝑘=1𝑑_𝑘𝜉_𝑘, and then ∀𝜀>0𝑃𝜇_𝑛𝐷_𝑛−𝐸𝜇_𝑛𝐷_𝑛≤𝜇≥𝜀Var_𝑛/𝐷_𝑛𝜀²≤𝑐log𝐷_𝑛^−𝛽. (A.1)
Let 𝑟<1, 𝑟𝛽>1, and 𝑛_𝑘=inf{𝑛_𝑘,𝐷_𝑛𝑘≥exp(𝑘^𝑟)}, and then 𝐷_𝑛𝑘≥exp(𝑘^𝑟),𝐷_𝑛𝑘−1<exp(𝑘^𝑟), for 𝐷_𝑛∼𝐷_𝑛−1; we get 𝐷1≤^𝑛𝑘exp𝑘^𝑟∼𝐷_𝑛𝑘−1exp𝑘^𝑟<1⟶1, (A.2) that is, 𝐷_𝑛𝑘∼exp𝑘^𝑟, (A.3) and thus 𝐷_𝑛𝑘𝐷_𝑛𝑘−1∼exp𝑘^𝑟exp(𝑘−1)^𝑟=exp𝑘^𝑟11−1−𝑘^𝑟∼exp𝑘^𝑟1⋅𝑟⋅𝑘=exp𝑟⋅𝑘^𝑟−1. (A.4) On account of 𝑟<1, then 𝐷_𝑛𝑘𝐷_𝑛𝑘−1∼exp𝑟⋅𝑘^𝑟−1⟶1,as𝑘⟶∞,_∞𝑘=1𝑃𝜇_𝑛𝑘𝐷_𝑛𝑘−𝐸𝜇_𝑛𝑘𝐷_𝑛𝑘≥𝜀≤𝑐^∞_𝑘=11log𝐷_𝑛^−𝛽≤𝑐^∞_𝑘=11𝑘^𝑟𝛽<∞. (A.5) By the Borel-Cantelli lemma, 𝜇_𝑛𝑘𝐷_𝑛𝑘−𝐸𝜇_𝑛𝑘𝐷_𝑛𝑘⟶0a.s. (A.6) Since 𝐸𝜇_𝑛𝑘𝐷_𝑛𝑘=∑^𝑛𝑘_𝑘=1𝑑_𝑘𝐷_𝑛𝑘=1, (A.7) thus 𝜇_𝑛𝑘𝐷_𝑛𝑘⟶1,a.s. (A.8) Now for 𝑛_𝑘−1≤𝑛<𝑛_𝑘, for 𝐷_𝑛↑∞,𝐷_𝑛𝑘/𝐷_𝑛𝑘−1→1, and by 𝜉_𝑘≥0, then 𝜇_𝑛↑, and we have 𝐷1⟵^𝑛𝑘−1𝐷_𝑛𝑘𝜇_𝑛𝑘−1𝐷_𝑛𝑘−1≤𝜇_𝑛𝐷_𝑛≤𝜇_𝑛𝑘𝐷_𝑛𝑘𝐷_𝑛𝑘𝐷_𝑛𝑘−1⟶1a.s. (A.9) hence 𝜇_𝑛𝐷_𝑛⟶1a.s. (A.10) This completes the proof of Lemma 6.

Proof of Lemma 10. Applying Lemma 9, (33), and noting the conditions of (19) and 0<𝛼<1/7, we get 𝑝_𝑘=𝑃𝑎_𝑘≤𝑆_𝑘≤𝑏_𝑘≤Φ𝑏_𝑘𝑘^1/2𝑎−Φ^𝑘𝑘^1/21+𝑐𝑘^1/5𝑘≤𝑐^1/2−𝛼𝑘^1/2+1𝑘^1/5≤𝑐^′₂𝑘^−𝛼. (A.11) Applying Lemma 9, (34), and noting the conditions of (19) and 0<𝛼<1/7, we have 𝑝_𝑘=𝑃𝑎_𝑘≤𝑆_𝑘≤𝑏_𝑘≥Φ𝑏_𝑘𝑘^1/2𝑎−Φ^𝑘𝑘^1/21−𝑐𝑘^1/5𝑘≥𝑐^1/2−𝛼𝑘^1/2+1𝑘^1/5≥𝑐^′₁𝑘^−𝛼. (A.12) Thus Lemma 10 immediately follows from (A.11) and (A.12).

Proof of Lemma 11. By Lemma 10, Chebyshev’s inequality, and noting the condition of 0<𝛼<1/7, we have _{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑘𝑙𝑝_𝑙𝑃𝑆_𝑙^𝛼≥𝜀_𝑙≤_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑘𝑙𝑝_𝑙Var𝑆_𝑙^𝛼𝑙^3𝛼≤𝑐^𝑛_𝑙=11𝑙^{1+𝛼𝑙−𝑙𝛼}_𝑘=11𝑘≤𝑐^𝑛_𝑙=1log𝑙−𝑙^𝛼𝑙^1+𝛼=𝑂log𝑛. (A.13) It proves (29). By Lemma 10 and 0<𝛼<1/7, we get _{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑘𝑙𝑝_𝑙1𝑙−𝑘−𝑙^𝛼1/5≤𝑐^𝑛_𝑙=11𝑙^1−𝛼_{1≤𝑘<(𝑙−𝑙}^𝛼_)/21𝑘𝑙−𝑙^𝛼−𝑘^1/5+_{(𝑙−𝑙𝛼)/2<𝑘<𝑙−𝑙𝛼}1𝑘𝑙−𝑙^𝛼−𝑘^1/5≤𝑐^𝑛_𝑙=11𝑙^1−𝛼1𝑙−𝑙^𝛼1/5_{1≤𝑘<(𝑙−𝑙}^𝛼_)/21𝑘+1𝑙−𝑙^𝛼_{1≤𝑘<(𝑙−𝑙𝛼)/2}1𝑘^1/5≤𝑐^𝑛_𝑙=1log𝑙𝑙^1−𝛼+1/5≤𝑐^𝑛_𝑙=11𝑙=𝑂log𝑛. (A.14) It proves (30). Applying Lemma 9, (33), (35), 𝜀_𝑙=𝑙^3𝛼/2, and noting the condition of 0<𝛼<1/7, we obtain _{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑘𝑙𝑝_𝑙Φ𝑎_𝑙−𝑏_𝑘−𝜀_𝑙𝑙−𝑘−𝑙^𝛼1/2𝑎−Φ^𝑙𝑙^1/2≤𝑐_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}−𝑎_𝑙𝑘𝑙𝑝_𝑙1𝑙−𝑘−𝑙^𝛼1/2−1√𝑙+𝑐_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑘𝑙𝑝_𝑙𝑏_𝑘𝑙−𝑘−𝑙^𝛼1/2+𝑐_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑘𝑙𝑝_𝑙𝜀_𝑙𝑙−𝑘−𝑙^𝛼1/2∶=Σ₁+Σ₂+Σ₃. (A.15) Now applying the same procedure as before, we have Σ₁≤𝑐_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}−𝑎_𝑙𝑘𝑙𝑝_𝑙1𝑙^1−𝛼𝑙−𝑘−𝑙^𝛼1/2+𝑘𝑙𝑙−𝑘−𝑙^𝛼1/2≤𝑐_{1≤𝑙≤𝑛}1𝑙^3/2−𝛼×1l−𝑙^𝛼1/2_{1≤𝑘<(𝑙−𝑙}^𝛼_)/21𝑘+1𝑙−𝑙^𝛼_{1≤𝑘<(𝑙−𝑙𝛼)/2}1𝑘^1/2+𝑐_{1≤𝑙≤𝑛}1𝑙^3/2_{1≤𝑘<𝑙−𝑙}^𝛼1𝑙−𝑘−𝑙^𝛼1/2≤𝑐_{1≤𝑙≤𝑛}log𝑙𝑙^2−𝛼≤𝑐_{1≤𝑙≤𝑛}1𝑙,Σ=𝑂log𝑛₂≤𝑐_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑙^1−𝛼𝑘^1/2+𝛼𝑙−𝑙^𝛼−𝑘^1/2≤𝑐_{1≤𝑙≤𝑛}1𝑙^1−𝛼1𝑙−𝑙^𝛼1/2_{1≤𝑘<(𝑙−𝑙}^𝛼_)/21𝑘^1/2+𝛼+1𝑙−𝑙^𝛼1/2+𝛼_{1≤𝑘<(𝑙−𝑙}^𝛼_)/21𝑘^1/2≤𝑐_{1≤𝑙≤𝑛}1𝑙^1−𝛼1𝑙−𝑙^𝛼1/2𝑙^1/2−𝛼+1𝑙−𝑙^𝛼1/2+𝛼𝑙−𝑙^𝛼1/2≤𝑐_{1≤𝑙≤𝑛}1𝑙=𝑂log𝑛. (A.16) Noting that 0<𝛼<1/7, we deduce Σ₃≤𝑐_{1≤𝑘<𝑙≤𝑛𝑘<𝑙−𝑙𝛼}1𝑙^{1−(7/2)𝛼}𝑘^1/2𝑙−𝑙^𝛼−𝑘^1/2≤𝑐_{1≤𝑙≤𝑛}1𝑙^{1−(7/2)𝛼}×1𝑙−𝑙^𝛼1/2_{1≤𝑘<𝑙−𝑙}^𝛼_/21𝑘+1𝑙−𝑙^𝛼_{1≤𝑘<𝑙−𝑙𝛼/2}1𝑘^1/2≤𝑐_{1≤𝑙≤𝑛}log𝑙𝑙^{3/2−(7/2)𝛼}≤𝑐_{1≤𝑙≤𝑛}1𝑙=𝑂log𝑛. (A.17) It proves (31). The proof of (32) is similar to the proof of (31). This completes the proof of Lemma 11.

Acknowledgments

The authors are very grateful to the academic editor, professor 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).