Weak and Almost Sure Convergence for Products of Sums of Associated Random Variables

Abstract

We study weak convergence of product of sums of stationary sequences of associated random variables to the log-normal law. The almost sure version of this result is also presented. The obtained theorems extend and generalize some of the results known so far for independent or associated random variables.

1. Introduction

In this paper, we consider sequences (𝑋_𝑛)_𝑛∈ℕ of random variables (r.v.) defined on some probability space (Ω,𝔉,𝑃), which are identically distributed but not necessarily independent. Let us assume that the r.v. are positive, square integrable and introduce the following notation 𝐸𝑋₁=𝜇, Var(𝑋₁)=𝜎²>0 and denote by 𝛾=𝜎/𝜇 the coefficient of variation, further let 𝑆_𝑛=∑^𝑛𝑘=1𝑋_𝑘. The first result concerning the asymptotic behavior of the products of partial sums of independent and identically distributed (i.i.d.) r.v. was obtained in 2002 by Rempała and Wesołowski (see [1]), who proved that∏^𝑛𝑖=1𝑆_𝑖𝑛!𝜇^𝑛^{√1/𝛾𝑛𝑑}⟶𝑒^√2𝒩,as𝑛⟶∞,(1.1) where 𝒩 is a standard normal variable, and →^𝑑 stands for the convergence in distribution.

Recently, this result was extended in different ways, we refer the reader to [2, 3], where further references are given. In particular Gonchigdanzan and Rempała (see [2]) studied the almost sure version of (1.1) and proved that for a sequence of i.i.d. r.v., we havelim_𝑛→∞1log𝑛^𝑛_𝑘=11𝑘𝐈⎡⎢⎢⎣∏^𝑘𝑗=1𝑆_𝑗𝑘!𝜇^𝑘^{√1/(𝛾𝑘)}≤𝑥⎤⎥⎥⎦=𝐹(𝑥),(1.2) almost surely (a.s.), for all 𝑥≥0. Here and in the sequel, 𝐈(⋅) is an indicator, and 𝐹(𝑥) is the distribution function of the log-normal random variable exp(√2𝒩).

Li and Wang (see [3]) extended the convergence in (1.2) to the case of associated r.v. Let us recall the notion of association introduced by Esary et al. in [4]. (𝑋_𝑛)_𝑛∈ℕ is a sequence of associated r.v., if for every finite subcollection 𝑋_𝑖1,…,𝑋_𝑖𝑛 and any coordinatewise nondecreasing functions 𝑓,𝑔∶ℝ^𝑛→ℝCov𝑓𝑋_𝑖1,…,𝑋_𝑖𝑛,𝑔𝑋_𝑖1,…,𝑋_𝑖𝑛≥0,(1.3) whenever this covariance exists. We will also use the notion of positively quadrant-dependent r.v. according to Lehmann (see [5]). The random variables 𝑋, 𝑌 are said to be positively quadrant dependent (PQD) if𝑃(𝑋≤𝑥,𝑌≤𝑦)≥𝑃(𝑋≤𝑥)𝑃(𝑌≤𝑦),(1.4) for all 𝑥,𝑦∈ℝ. We refer the reader to the monograph [6] of Bulinski and Shashkin as the survey on association. Let us only mention that associated r.v. are pairwise PQD, independent r.v. are associated, and associated and uncorrelated r.v. are independent. Moreover, nondecreasing functions of independent r.v. and nonnegatively correlated Gaussian r.v. are associated. Associated and PQD r.v. are examples of the so-called positively dependent r.v. They are nonnegatively correlated, and in view of the properties, the covariance is often used as a measure of their dependence.

The motivation for the study of the convergence in (1.1) goes back to the paper of Arnold and Villaseñor (see [7]), who considered the limiting properties of sums of record values. The convergence in (1.1) may be also used, for example, to construct the distribution-free tests for the coefficient of variation 𝛾. On the other hand, associated r.v. play very important role in mathematical physics and statistics, and there is a lot of papers devoted to limit theorems under this kind of dependence (see [6, 8, 9]).

The aim of our paper is to generalize the results of [1, 2] by proving (1.1) and (1.2) for strictly stationary sequences of associated r.v. We also relax the condition on the rate of decay of the covariance coefficients imposed in [3].

2. Main Results

Let us state the first of our main results. In the following theorem, we generalize the main result of [1]. We shall consider sequences of associated r.v. instead of independent r.v.

Theorem 2.1. Let (𝑋_𝑛)_𝑛∈ℕ be a strictly stationary sequence of positive and square-integrable associated r.v. Assume that 0<𝜎²¹∶=𝜎²+2^∞_𝑘=1Cov𝑋₁,𝑋_𝑘+1<∞.(2.1) Then, ∏^𝑛𝑘=1𝑆_𝑘𝑛!𝜇^𝑛^{1/𝛾1√𝑛𝑑}⟶𝑒^√2𝒩,as𝑛⟶∞,(2.2) where 𝛾₁=𝜎₁/𝜇.

Remark 2.2. Let us note that i.i.d. sequences are trivial examples of stationary associated sequences satisfying (2.1). Therefore, the above theorem is a generalization of the main result in [1]. Furthermore, our proof simplifies the original one by employing the Marcinkiewicz-Zygmund-type theorem. As a nontrivial example of associated r.v. satisfying our assumptions, one can take a stationary sequence of Gaussian r.v. with covariance structure such that (2.1) holds. Other examples may be constructed, for example, as the moving averages of i.i.d. r.v. with positive coefficients.

In the next theorem, we present the almost sure version of the limiting result obtained in Theorem 2.1. Our results generalize the ones obtained in [2, 3].

Theorem 2.3. Assume that the conditions of Theorem 2.1 are satisfied, then, lim_𝑛→∞sup_𝑥≥0||||||1log𝑛^𝑛_𝑖=11𝑖𝐈⎡⎢⎢⎣∏^𝑖𝑘=1𝑆_𝑘𝑖!𝜇^𝑖^{1/𝛾1√𝑖}≤𝑥⎤⎥⎥⎦−𝐹(𝑥)||||||=0a.s.,(2.3) where (2.3) 𝐹(𝑥) denotes the distribution of the random variable (2.3) exp(√2𝒩).

Remark 2.4. Our Theorem 2.3 generalizes the result of [2], where i.i.d. r.v. were considered. It is also more general than the main result of [3], where a stronger and technical condition Cov𝑋₁,𝑋_𝑛+1=𝑂𝑛⁻¹(log𝑛)^−2−𝜀,forsome𝜀>0(2.4) was imposed. Our assumption (2.1) on the summability of covariances is natural and was used previously in the literature (see, e.g., [9]). Let us also mention that the method used in the proof is different than in [2, 3].

3. Auxiliary Lemmas and Proofs of the Main Results

In order to prove Theorem 2.1, we need four lemmas. For the first one, let us introduce the following notation, which will be also used in the sequel. Let us denote by 𝜎^2𝑛 the variance of the sum ∑^𝑛𝑘=1((𝑆_𝑘−𝑘𝜇)/𝑘), then,𝜎^2𝑛=Var^𝑛_𝑘=1𝑏_𝑘,𝑛𝑋_𝑘−𝜇=Var^𝑛_𝑘=1𝑏_𝑘,𝑛𝑋_𝑘,(3.1) where 𝑏_𝑘,𝑛=∑^{𝑛𝑖=𝑘}(1/𝑖).

The first lemma describes the asymptotic behavior of the variance 𝜎^2𝑛.

Lemma 3.1. Assume that the conditions of Theorem 2.1 are satisfied, then, lim_𝑛→∞𝜎^2𝑛2𝑛=𝜎²¹=𝜎²+2^∞_𝑘=1Cov𝑋₁,𝑋_𝑘+1.(3.2)

Proof. Let us observe that, by stationarity, we have 𝜎^2𝑛2𝑛=∑^𝑛𝑘=1𝑏^2𝑘,𝑛Var𝑋_𝑘2𝑛+2∑^{𝑛−1𝑘=1}𝑓_𝑘,𝑛Cov𝑋₁,𝑋_𝑘+12𝑛,(3.3) where 𝑓_𝑘,𝑛=𝑏_1,𝑛𝑏_1+𝑘,𝑛+𝑏_2,𝑛𝑏_2+𝑘,𝑛+⋯+𝑏_{𝑛−𝑘,𝑛}𝑏_𝑛,𝑛=^𝑛−𝑘_𝑗=1𝑏_𝑗,𝑛𝑏_{𝑗+𝑘,𝑛}.(3.4) For 𝑘∈ℕ, let us put 𝑔_𝑘=∑^∞𝑖=𝑘Cov(𝑋₁,𝑋_𝑖+1), then 𝑔_𝑘+1−𝑔_𝑘=−Cov(𝑋₁,𝑋_𝑘+1). In order to calculate the second summand in (3.3), we shall apply the summation by parts formula ^𝑛−1_𝑘=1𝑓_𝑘,𝑛𝑔_𝑘+1−𝑔_𝑘=𝑔_𝑛𝑓_{𝑛−1,𝑛}−𝑓_1,𝑛𝑔₁−^𝑛−2_𝑘=1𝑔_𝑘+1𝑓_𝑘+1,𝑛−𝑓_𝑘,𝑛.(3.5)
Thus, in our case, ^𝑛−1_𝑘=1𝑓_𝑘,𝑛Cov𝑋₁,𝑋_𝑘+1=𝑓_1,𝑛𝑔₁−𝑔_𝑛𝑓_{𝑛−1,𝑛}+^𝑛−2_𝑘=1𝑔_𝑘+1𝑓_𝑘+1,𝑛−𝑓_𝑘,𝑛.(3.6) Applying the identity ^𝑛_𝑗=1𝑏^2𝑗,𝑛=2𝑛−𝑏_1,𝑛(3.7) (see formula (4) in [1]), we get 𝑓_1,𝑛=^𝑛−1_𝑗=1𝑏_𝑗,𝑛𝑏_𝑗+1,𝑛=^𝑛−1_𝑗=1𝑏_𝑗,𝑛𝑏^𝑗,𝑛−1=_𝑛−1𝑗_𝑗=1𝑏^2𝑗,𝑛−^𝑛−1_𝑗=1𝑏_𝑗,𝑛1𝑗=2𝑛−𝑏^1,𝑛−1𝑛²−^𝑛−1_𝑗=1𝑏_𝑗,𝑛1𝑗.(3.8)
Thus, taking into account that 𝑓_{𝑛−1,𝑛}=𝑏_1,𝑛𝑏_𝑛,𝑛, the formula (3.3) takes the following form: 𝜎^2𝑛2𝑛=𝜎²∑^𝑛𝑘=1𝑏^2𝑘,𝑛2𝑛+2𝑓^1,𝑛𝑔₁−𝑔_𝑛𝑓_{𝑛−1,𝑛}+∑^{𝑛−2𝑘=1}𝑔_𝑘+1𝑓_𝑘+1,𝑛−𝑓_𝑘,𝑛=𝜎₂2𝑛2𝑛−2𝑛𝑏^1,𝑛+2𝑛−𝑏^1,𝑛−1/𝑛²−∑^{𝑛−1𝑗=1}𝑏_𝑗,𝑛(1/𝑗)𝑛^∞_𝑘=1−∑_∞𝑖=𝑛Cov𝑋₁,𝑋_𝑘+1Cov𝑋₁,𝑋_𝑖+1𝑏_1,𝑛𝑏_𝑛,𝑛𝑛−𝑓^1,𝑛−𝑓_{𝑛−1,𝑛}𝑛^𝑛−2_𝑘=1𝑔_𝑘+1𝑐_{𝑘,𝑛−2},(3.9) where 𝑐_{𝑘,𝑛−2}, for 𝑘=1,2,…,𝑛−2 are defined as follows: 𝑐_{𝑘,𝑛−2}=𝑓^𝑘,𝑛−𝑓_𝑘+1,𝑛∑^{𝑛−2𝑘=1}𝑓_𝑘,𝑛−𝑓_𝑘+1,𝑛=𝑓^𝑘,𝑛−𝑓_𝑘+1,𝑛𝑓_1,𝑛−𝑓_{𝑛−1,𝑛}.(3.10) On account that 𝑓_𝑘,𝑛>𝑓_𝑘+1,𝑛, we have 𝑐_{𝑘,𝑛−2}>0. Furthermore, ∑^{𝑛−2𝑘=1}𝑐_{𝑘,𝑛−2}=1. We shall prove that 𝑐_{𝑘,𝑛−2}→0, as 𝑛→∞, for every 𝑘∈ℕ. At first, let us note that 0<𝑓_𝑘,𝑛−𝑓_𝑘+1,𝑛=^𝑛−𝑘_𝑗=1𝑏_𝑗,𝑛𝑏_{𝑗+𝑘,𝑛}−^{(𝑛−𝑘)−1}_𝑗=1𝑏_𝑗,𝑛𝑏_{𝑗+𝑘+1,𝑛}=𝑏_{𝑛−𝑘,𝑛}𝑏_𝑛,𝑛+^{(𝑛−𝑘)−1}_𝑗=1𝑏_𝑗,𝑛1𝑗+𝑘=𝑏^{𝑛−𝑘,𝑛}𝑏_𝑛,𝑛+^{(𝑛−𝑘)−1}_𝑛𝑗=1_𝑖=𝑗1<𝑏_1,𝑛𝑖𝑗+1𝑘𝑏_𝑛,𝑛+^𝑛_𝑛𝑗=1_𝑖=11𝑖1𝑗=𝑏^1,𝑛𝑏_𝑛,𝑛+𝑏_1,𝑛²<2(1+log𝑛)²,(3.11) by the inequality 𝑏_1,𝑛−log𝑛<1. On the other hand, we have 𝑓_1,𝑛−𝑓_{𝑛−1,𝑛}=^𝑛−1_𝑗=1𝑏_𝑗,𝑛𝑏_𝑗+1,𝑛−𝑏_1,𝑛𝑏_𝑛,𝑛=^𝑛−1_𝑗=1𝑏_𝑗,𝑛𝑏^𝑗,𝑛−1𝑗−𝑏^1,𝑛𝑏_𝑛,𝑛=^𝑛_𝑗=1𝑏^2𝑗,𝑛−𝑏^2𝑛,𝑛−^𝑛−1_𝑗=1𝑏_𝑗,𝑛1𝑗−𝑏^1,𝑛𝑏_𝑛,𝑛=2𝑛−𝑏_1,𝑛−1𝑛²−^𝑛−1_𝑗=1𝑏_𝑗,𝑛1𝑗−𝑏^1,𝑛1𝑛>2𝑛−(2+log𝑛)²>𝑛,(3.12) for sufficiently large 𝑛 (𝑛≥29). Therefore, from (3.11) and (3.12), we get 𝑐_{𝑘,𝑛−2}≤2(1+log𝑛)²𝑛⟶0,(3.13) for each 𝑘∈ℕ. From the Toeplitz theorem on the transformation of sequences into sequences (Problem  2.3.1 in [10]), we obtain ^𝑛−2_𝑘=1𝑔_𝑘+1𝑐_{𝑘,𝑛−2}⟶0,(3.14) since 𝑔_𝑘+1→0. It is easy to see that 0<𝑓_1,𝑛−𝑓_{𝑛−1,𝑛}<∑^𝑛𝑗=1𝑏^2𝑗,𝑛<2𝑛.
Thus, we see that from (3.9) the conclusion follows.

In the following lemma let us recall Theorem 2.3from [11], which will be one of the tools needed in the sequel.

Lemma 3.2. Let {𝑎_𝑛,𝑘;𝑛∈ℕ,1≤𝑘≤𝑛} be a triangular array of real numbers satisfying sup_{𝑛𝑛∈ℕ}_𝑘=1𝑎^2𝑛,𝑘<∞,lim_𝑛→∞max_{1≤𝑘≤𝑛}||𝑎_𝑛,𝑘||=0.(3.15) Let (𝑌_𝑛)_𝑛∈ℕ be a sequence of centered, square-integrable associated r.v. such that the family {𝑌^2𝑛;𝑛∈ℕ} is uniformly integrable and Var^𝑛_𝑘=1𝑎_𝑛,𝑘𝑌_𝑘=1.(3.16)
If furthermore ∑_{𝑗∶|𝑘−𝑗|≥𝑛}Cov(𝑌_𝑘,𝑌_𝑗)→0, uniformly in 𝑘∈ℕ, as 𝑛→∞, then ∑^𝑛𝑘=1𝑎_𝑛,𝑘𝑌→_𝑘𝑑𝒩, as 𝑛→∞.

In the next lemma, we state the central limit theorem for sums of associated r.v. The proof particularly relies on Lemmas 3.1 and 3.2.

Lemma 3.3. Assume that the conditions of Theorem 2.1 are satisfied, then 1√2𝑛𝜎_1𝑛𝑘=1𝑆_𝑘−𝑘𝜇𝑘=√21𝑛𝜎_1𝑛𝑘=1𝑏_𝑘,𝑛𝑋_𝑘−𝜇⟶^𝑑𝒩,as𝑛⟶∞.(3.17)

Proof. We apply Lemma 3.2 to the sequence 𝑌_𝑘=𝑋_𝑘−𝜇 with 𝑎_𝑛,𝑘=𝑏_𝑘,𝑛/𝜎_𝑛. Let us check the assumptions. Obviously the family {𝑌^2𝑛;𝑛∈ℕ} is uniformly integrable. It is easy to see that by Lemma 3.1^𝑛_𝑘=1𝑎^2𝑛,𝑘=∑^𝑛𝑘=1𝑏^2𝑘,𝑛𝜎^2𝑛=2𝑛−𝑏^1,𝑛2𝑛2𝜎𝑛^2𝑛⟶1𝜎²¹,as𝑛⟶∞,(3.18) thus, sup_𝑛∈ℕ∑^𝑛𝑘=1𝑎^2𝑛,𝑘<∞. Further we have max_{1≤𝑘≤𝑛}|𝑎_𝑛𝑘|=𝑏_1,𝑛/𝜎_𝑛≤(1+log𝑛)/𝑐√𝑛→0. Obviously Var(∑^𝑛𝑘=1𝑎_𝑛,𝑘𝑌_𝑘)=1, thus, 1𝜎_{𝑛𝑛𝑘=1}1𝑏_𝑘,𝑛𝑋_𝑘−𝜇⟶^𝑑𝒩,√2𝑛𝜎_1𝑛𝑘=1𝑏_𝑘,𝑛𝑋_𝑘−𝜇=1√2𝑛𝜎₁𝜎_𝑛1𝜎_{𝑛𝑛𝑘=1}𝑏_𝑘,𝑛𝑋_𝑘−𝜇⟶^𝑑𝒩,(3.19) since we have 𝜎^2𝑛/2𝑛𝜎²¹→1,as𝑛→∞.

We shall also need the following strong law of large numbers of the Marcinkiewicz-Zygmund type.

Lemma 3.4. Let (𝑋_𝑛)_𝑛∈ℕ be a strictly stationary sequence of square-integrable associated r.v. If ∑^∞𝑗=2Cov(𝑋₁,𝑋_𝑗)<∞, then for every 𝑝∈(0,2), one has 1𝑛^𝑛1/𝑝_𝑘=1𝑋_𝑘−𝐸𝑋_𝑘⟶0,almostsurely,as𝑛⟶∞.(3.20)

Proof. We directly apply Theorem  3.3 in [12]. This result states that if (𝑏_𝑛)_𝑛∈ℕ is a nondecreasing sequence of positive real numbers and (𝑋_𝑛)_𝑛∈ℕ is a sequence of associated r.v. with ^∞_𝑗=1Var𝑋_𝑗𝑏^2𝑗+^∞_𝑗_≠𝑘Cov𝑋_𝑗,𝑋_𝑘𝑏_𝑗𝑏_𝑘<∞,(3.21) then, 𝑏^𝑛−1∑^𝑛𝑗=1(𝑋_𝑗−𝐸𝑋_𝑗)→0 almost surely, as 𝑛→∞.

Proof of Theorem 2.1. After taking the logarithm it suffices to show that 𝐿∶=1𝛾₁√2𝑛^𝑛_𝑘=1log𝑆^𝑘𝑘𝜇^𝑑⟶𝒩.(3.22) We shall use the expansion of the logarithm log(1+𝑥)=𝑥−(𝑥²/2(1+𝜃𝑥)²), where 𝜃∈(0,1) depends on 𝑥>−1. We have 𝐿=1𝛾₁√2𝑛^𝑛_𝑘=1𝑆_𝑘−𝑘𝜇𝑘𝜇−𝛾₁√12𝑛^𝑛_𝑘=1𝑆_𝑘−𝑘𝜇/𝑘𝜇²21+𝜃_𝑘𝑆_𝑘−𝑘𝜇/𝑘𝜇²,(3.23) where 𝜃_𝑘 are random variables taking values in (0,1). The first term in (3.23) converges weakly to the standard normal law by Lemma 3.3. We shall prove that the second one converges almost surely to 0. By Lemma 3.4 with 𝑝=4/3, we get (𝑆_𝑘−𝑘𝜇)/𝑘^3/4)→0,almostsurelyas𝑘→∞, consequently (𝑆_𝑘−𝑘𝜇)/𝑘→0 almost surely, as 𝑘→∞. Therefore, 𝑆_𝑘−𝑘𝜇/𝑘^3/4²1+𝜃_𝑘𝑆_𝑘−𝑘𝜇/𝑘𝜇²⟶0,almostsurely,as𝑘→∞.(3.24) The weighted average with weights 1/√𝑘 is also convergent to 0, accordingly 1𝛾₁√2𝑛^𝑛_𝑘=1𝑆_𝑘−𝑘𝜇/𝑘𝜇²1+𝜃_𝑘𝑆_𝑘−𝑘𝜇/𝑘𝜇²=1𝛾₁√2𝑛^𝑛_𝑘=11√𝑘𝑆^𝑘−𝑘𝜇/𝑘^3/4𝜇²1+𝜃_𝑘𝑆_𝑘−𝑘𝜇/𝑘𝜇²⟶0,(3.25) almost surely, as 𝑛→∞, and the proof is completed.

In the proof of Theorem 2.3, we shall use the following inequality concerning indicators. The proof is immediate.

Lemma 3.5. For any random variables 𝑋, 𝑌 and real numbers 𝑡∈ℝ, 𝑎>0, the following inequalities hold: 𝐈𝐈[[𝑋𝑋++𝑌𝑌≤≤𝑡𝑡]]≤≥𝐈𝐈[[𝑋𝑋≤≤𝑡𝑡+−𝑎𝑎]]+−𝐈𝐈||||𝑌𝑌||||>>𝑎𝑎,.(3.26) The proof of Theorem 2.3 is based on Theorem  1 in [8], let us state it here as a lemma.

Lemma 3.6. Let (𝑎_𝑛)_𝑛∈ℕ be a sequence of positive numbers, one puts 𝑏_𝑛=∑^𝑛𝑘=1𝑎_𝑘, and assume that 𝑎_𝑛/𝑏_𝑛→0,𝑏_𝑛→∞ as 𝑛→∞. Let (𝑌_𝑛)_𝑛∈ℕ be a sequence of pairwise PQD r.v. with distribution functions 𝐹_𝑛(𝑥), respectively. Suppose that 𝐹→_𝑛𝑑𝐹, where 𝐹 is a continuous distribution function, and ^∞_𝑗𝑗=1_𝑖=1𝑎_𝑖𝑎_𝑗Cov𝑌_𝑖,𝑌_𝑗𝑏^2𝑗<∞.(3.27) Then, sup_{−∞<𝑥<∞}|||||1𝑏_{𝑛𝑛𝑖=1}𝑎_𝑖𝐈𝑌_𝑖≤𝑥−𝐹(𝑥)|||||⟶0a.s.,as𝑛⟶∞.(3.28)

Proof of Theorem 2.3. Let 𝜀>0 be fixed. We put 𝐶_𝑘=𝑆_𝑘/𝑘𝜇 and log𝑥=𝑡. By using the expansion of the logarithm and Lemma 3.5, we get 𝐈⎡⎢⎢⎣∏^𝑖𝑘=1𝑆_𝑘𝑖!𝜇^𝑖^{1/𝛾1√𝑖}≤𝑥⎤⎥⎥⎦=𝐈1𝛾₁√𝑖^𝑖_𝑘=1𝐶_𝑘≤𝐈1−1+𝑅_𝑖≤𝑡𝛾₁√𝑖^𝑖_𝑘=1𝐶_𝑘𝐈⎡⎢⎢⎣∏−1≤𝑡+𝜀+𝐈||𝑅_𝑖||>𝜀_𝑖𝑘=1,𝑆_𝑘𝑖!𝜇^𝑖^{1/𝛾1√𝑖}≤𝑥⎤⎥⎥⎦≥𝐈1𝛾₁√𝑖^𝑖_𝑘=1𝐶_𝑘−1≤𝑡−𝜀−𝐈||𝑅_𝑖||>𝜀,(3.29) where 𝑅_𝑖=(−1/𝛾₁√𝑖)∑^𝑖𝑘=1((𝐶_𝑘−1)²/2(1+𝜃_𝑘(𝐶_𝑘−1))²). As in the proof of Theorem 2.1, we see that 𝑅_𝑖→0 almost surely, as 𝑖→∞. Thus, for almost every 𝜔∈Ω there exists 𝑖₀(𝜔) such that for 𝑖≥𝑖₀ we have |𝑅_𝑖(𝜔)|<𝜀, therefore, 𝐈||𝑅_𝑖(𝜔)||>𝜀⟶0,𝑖⟶∞,(3.30) for almost every 𝜔. Now, we shall apply Lemma 3.6 to the sequence 𝑌_𝑖=1𝛾₁√𝑖^𝑖_𝑘=1𝐶_𝑘−1,(3.31) and weights 𝑎_𝑖=1/𝑖 and 𝑏_𝑛=log𝑛. The r.v. 𝑌_𝑖 are not only PQD but associated as well, as sums of the r.v. 𝐶_𝑘−1 which are associated. By Lemma 3.3, we have 1𝛾₁√𝑖^𝑖_𝑘=1𝐶_𝑘−1⟶^𝑑√2𝒩,as𝑖⟶∞.(3.32) We shall check the condition (3.27). Let us observe that 𝑌_𝑖=1𝛾₁√𝑖^𝑖_𝑘=1𝐶_𝑘−1=1𝜎₁√𝑖^𝑖_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇,(3.33) whereas before 𝑏_𝑘,𝑖=∑^{𝑖𝜈=𝑘}(1/𝜈).
For 1≤𝑖<𝑗, we have the following decomposition: ^𝑗_𝑘=1𝑏_𝑘,𝑗𝑋_𝑘−𝜇=^𝑖_𝑘=1𝑏_𝑘,𝑖+𝑏_𝑖+1,𝑗𝑋_𝑘−𝜇+^𝑗_𝑘_=𝑖+1=𝑏_𝑘,𝑗𝑋_𝑘−𝜇_𝑖_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇+𝑏_{𝑖𝑖+1,𝑗}_𝑘=1𝑋_𝑘−𝜇+^𝑗_𝑘_=𝑖+1𝑏_𝑘,𝑗𝑋_𝑘−𝜇.(3.34) Thus, for 1≤𝑖<𝑗, we get 0≤Cov𝑌_𝑖,𝑌_𝑗=1𝜎²¹√𝑖𝑗Var^𝑖_𝑘=1+Cov_𝑖𝑏_𝑘,𝑖𝑋_𝑘−𝜇_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇;𝑏_{𝑖𝑖+1,𝑗}_𝑘=1𝑋_𝑘+Cov_𝑖−𝜇_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇;^𝑗_𝑘_=𝑖+1𝑏_𝑘,𝑗𝑋_𝑘−𝜇.(3.35) We shall find the bounds for each of the above terms. By Lemma 3.1, we have Var^𝑖_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇=𝜎^2𝑖≤𝐶₁𝑖,(3.36) for some constant 𝐶₁>0 and every 𝑖∈ℕ. From the Cauchy-Schwarz inequality, we get Cov^𝑖_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇;𝑏_{𝑖𝑖+1,𝑗}_𝑘=1𝑋_𝑘≤⎷−𝜇Var^𝑖_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇⋅𝑏_𝑖+1,𝑗⎷Var^𝑖_𝑘=1𝑋_𝑘−𝜇.(3.37) Stationarity, association and our assumption (2.1) imply lim_𝑖→∞1𝑖Var^𝑖_𝑘=1𝑋_𝑘−𝜇=𝜎²¹,(3.38) thus, Var^𝑖_𝑘=1𝑋_𝑘−𝜇≤𝐶₂𝑖,(3.39) for some constant 𝐶₂>0 and every 𝑖∈ℕ. Furthermore, 𝑖+1≤𝑗, thus, 𝑏_𝑖+1,𝑗=^𝑗_𝜈=11𝜈−^𝑖_𝜈=11𝜈≤1+log𝑗−𝛾^𝐸≤1+log𝑗−log𝑖𝑖≤𝑝𝑗𝑖^1/𝑝,(3.40) what follows from the inequalities 𝛾_𝐸≤^𝑛_𝜈=111+log𝑥<𝑝𝑥𝜈_1/𝑝−log𝑛≤1,,𝑝>1,𝑥≥1,(3.41) where 𝛾_𝐸=lim_𝑛→∞(∑^𝑛𝑘=1(1/𝑘)−log𝑛) denotes the Euler constant. From (3.36)–(3.40), we get Cov^𝑖_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇;𝑏_{𝑖𝑖+1,𝑗}_𝑘=1𝑋_𝑘−𝜇≤√𝐶₁𝐶₂⋅𝑝⋅𝑖⋅𝑗𝑖^1/𝑝.(3.42) Now, let us estimate the last term in (3.35): 𝐴∶=Cov^𝑖_𝑘=1𝑏_𝑘,𝑖𝑋_𝑘−𝜇;^𝑗_𝑘_=𝑖+1𝑏_𝑘,𝑗𝑋_𝑘−𝜇=𝐴₁+𝐴₂,say,(3.43) where 𝐴₁=^𝑖_{𝜈=2min(𝑖−1,𝑗−𝑖)𝑠=1}𝐴₂Cov𝑋_{𝜈+𝑠−1};𝑋_𝑖+𝑠𝑏_{𝜈+𝑠−1,𝑖}𝑏_{𝑖+𝑠,𝑗},=^𝑗−𝑖_{𝜈=1min(𝑖,𝑗−𝑖)𝑠=1}Cov𝑋_𝑠;𝑋_{𝜈+𝑠+𝑖−1}𝑏_𝑠,𝑖𝑏_{𝜈+𝑠+𝑖−1,𝑗},(3.44) for 𝑖=1, we put 𝐴₁=0.
By the monotonicity properties of the coefficients 𝑏_𝑛,𝑚 with respect to 𝑛 and fixed 𝑚, Cauchy-Schwarz inequality, (3.40), and formula (3.7), we get ^min(^{𝑖−1,𝑗−𝑖)}_𝑠=1𝑏_{𝜈+𝑠−1,𝑖}𝑏_{𝑖+𝑠,𝑗}≤^{min(𝑖−1,𝑗−𝑖)}_𝑠=1𝑏_𝑠,𝑖𝑏_{𝑖+𝑠,𝑗}≤⎷^{min(𝑖−1,𝑗−𝑖)}_𝑠=1𝑏^2𝑠,𝑖⋅⎷^{min(𝑖−1,𝑗−𝑖)}_𝑠=1𝑏^{2𝑖+𝑠,𝑗}≤⎷^𝑖_𝑠=1𝑏^2𝑠,𝑖⋅⎷^{min(𝑖−1,𝑗−𝑖)}_𝑠=1𝑏^2𝑖+1,𝑗≤𝑝√2⋅𝑖⋅𝑗𝑖^1/𝑝,(3.45) and by stationarity 𝐴₁≤𝑝√2⋅𝑖⋅𝑗𝑖^𝑖1/𝑝_𝜈=2=𝑝√Cov𝑋_𝑖+1;𝑋_𝜈2⋅𝑖⋅𝑗𝑖^𝑖1/𝑝_𝜈=2Cov𝑋₁;𝑋_𝜈.(3.46) Similarly, we deal with 𝐴₂. We have ^min^{(𝑖,𝑗−𝑖)}_𝑠=1𝑏_𝑠,𝑖𝑏_{𝜈+𝑠+𝑖−1,𝑗}≤⎷^{min(𝑖,𝑗−𝑖)}_𝑠=1𝑏^2𝑠,𝑖⋅⎷^{min(𝑖,𝑗−𝑖)}_𝑠=1𝑏^{2𝜈+𝑠+𝑖−1,𝑗}≤⎷^𝑖_𝑠=1𝑏^2𝑠,𝑖⋅⎷^𝑖_𝑠=1𝑏^2𝑖+1,𝑗≤𝑝√2⋅𝑖⋅𝑗𝑖^1/𝑝,(3.47) and by stationarity 𝐴₂≤𝑝√2⋅𝑖⋅𝑗𝑖^{1/𝑝𝑗−𝑖}_𝜈=1Cov𝑋₁;𝑋_𝜈+𝑖.(3.48) Therefore, from (3.46), (3.48), and (2.1), we get 𝐴≤𝑝√2⋅𝑖⋅𝑗𝑖^𝑗1/𝑝_𝜈=2Cov𝑋₁;𝑋_𝜈≤𝜎²¹𝑝√2⋅𝑖⋅𝑗𝑖^1/𝑝.(3.49) Clearly, Cov(𝑌_𝑖,𝑌_𝑖) is bounded by some constant for all 𝑖≥1. So that from (3.35), (3.36), (3.42) and (3.49), we obtain Cov𝑌_𝑖,𝑌_𝑗≤𝐶₃𝑗𝑖^1/2−1/𝑝,(3.50) for all 1≤𝑖≤𝑗 and 𝑝>1, where the constant 𝐶₃>0 is independent of 𝑖 and 𝑗 but may depend on 𝑝. Now, we shall check that the condition (3.27) holds. By (3.50) with 𝑝>2, we have ^∞_𝑗𝑗=1_𝑖=11𝑖𝑗log²𝑗Cov𝑌^𝑖,𝑌_𝑗≤𝐶_3∞_𝑗=11𝑗log²𝑗𝑗1^{𝑗(1/2)−(1/𝑝)}_𝑖=11𝑖𝑖^{(1/2)−(1/𝑝)}≤𝐶_4∞_𝑗=11𝑗log²𝑗𝑗1^{(1/2)−(1/𝑝)}𝑗^{(1/2)−(1/𝑝)}<∞,(3.51) for some 𝐶₄>0, since ∑^𝑗𝑖=1𝑖^{−(1/2)−(1/𝑝)} is of order 𝑗^{(1/2)−(1/𝑝)}. By (3.28), we get 1log𝑛^𝑛_𝑖=11𝑖𝐈𝑌^𝑖≤𝑡+𝜀⟶Φ𝑡+𝜀1√2a.s.,as𝑛⟶∞,log𝑛^𝑛_𝑖=11𝑖𝐈𝑌^𝑖≤𝑡−𝜀⟶Φ𝑡−𝜀√2a.s.,as𝑛⟶∞,(3.52) where Φ(𝑥) is the standard normal distribution. By the inequality |Φ(𝑡)−Φ(𝑡+𝑎)|≤𝑎/√2𝜋, we get Φ𝑡+𝜀√2<Φ√𝑡2+𝜀,Φ𝑡√−2𝜀>Φ√𝑡2−𝜀.(3.53) Combining the above considerations, we get for every 𝜀>0limsup_𝑛→∞1log𝑛^𝑛_𝑖=11𝑖𝐈⎡⎢⎢⎣∏^𝑖𝑘=1𝑆_𝑘𝑖!𝜇^𝑖^{1/𝛾1√𝑖}≤𝑥⎤⎥⎥⎦≤Φ𝑡liminf_𝑛→∞√2+𝜀a.s.,1log𝑛^𝑛_𝑖=11𝑖𝐈⎡⎢⎢⎣∏^𝑖𝑘=1𝑆_𝑘𝑖!𝜇^𝑖^{1/𝛾1√𝑖}≤𝑥⎤⎥⎥⎦≥Φ𝑡√2−𝜀a.s.,(3.54) since 𝜀>0 was arbitrary, we conclude that lim_𝑛→∞1log𝑛^𝑛_𝑖=11𝑖𝐈⎡⎢⎢⎣∏^𝑖𝑘=1𝑆_𝑘𝑖!𝜇^𝑖^{1/𝛾1√𝑖}≤𝑥⎤⎥⎥⎦=Φlog𝑥√2a.s.(3.55) But Φ(log𝑥/√2)=𝐹(𝑥) is the distribution of exp(√2𝒩). By continuity of 𝐹 and the same arguments as in [8], we get the uniform convergence and the conclusion follows.

Acknowledgement

The authors would like to thank the referees for careful reading of the paper and helpful comments.