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

In this paper, we consider sequences Xn 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 Sn ∑n k 1 Xk. 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


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 EX 1 μ, Var X 1 σ 2 > 0 and denote by γ σ/μ the coefficient of variation, further let S n n k 1 X 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 where N is a standard normal variable, and d → stands for the convergence in distribution.

ISRN Probability and Statistics
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 have 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 i 1 , . . ., X i n and any coordinatewise nondecreasing functions f, g : 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 if 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 Villase ñor 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 distributionfree 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 .

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 where F x denotes the distribution of the random variable 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 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 .

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 2 n the variance of the sum n k 1 S k − kμ /k , then, where b k,n n i k 1/i .The first lemma describes the asymptotic behavior of the variance σ 2 n .
Lemma 3.1.Assume that the conditions of Theorem 2.1 are satisfied, then, Proof.Let us observe that, by stationarity, we have where In order to calculate the second summand in 3.3 , we shall apply the summation by parts formula Thus, in our case, Applying the identity

3.8
Thus, taking into account that f n−1,n b 1,n b n,n , the formula 3.3 takes the following form: 3.9 where c k,n−2 , for k 1, 2, . . ., n − 2 are defined as follows:

3.11
by the inequality b 1,n − log n < 1.On the other hand, we have 3.12 for sufficiently large n n ≥ 29 .Therefore, from 3.11 and 3.12 , we get for each k ∈ N. From the Toeplitz theorem on the transformation of sequences into sequences Problem 2.3.1 in 10 , we obtain . Thus, we see that from 3.9 the conclusion follows.
In the following lemma let us recall Theorem 2.3 from 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

3.15
Let Y n n∈N be a sequence of centered, square-integrable associated r.v.such that the family {Y 2 n ; n ∈ N} is uniformly integrable and 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
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 2 n ; n ∈ N} is uniformly integrable.It is easy to see that by Lemma 3.1 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 2 Cov X 1 , X j < ∞, then for every p ∈ 0, 2 , one has

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 Proof of Theorem 2.1.After taking the logarithm it suffices to show that

3.24
The weighted average with weights 1/ √ k is also convergent to 0, accordingly 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:

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 n k 1 a 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 n d → F, where F is a continuous distribution function, and

3.28
Proof of Theorem 2.3.Let ε > 0 be fixed.We put C k S k /kμ and log x t.By using the expansion of the logarithm and Lemma 3.5, we get 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 i 0 ω such that for i ≥ i 0 we have |R i ω | < ε, therefore, for almost every ω.Now, we shall apply Lemma 3.6 to the sequence and weights a i 1/i and b n log n.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

ISRN Probability and Statistics
We shall check the condition 3.27 .Let us observe that whereas before b k,i i ν k 1/ν .For 1 ≤ i < j, we have the following decomposition:

3.34
Thus, for 1 ≤ i < j, we get

3.35
We shall find the bounds for each of the above terms.By Lemma 3.1, we have for some constant C 1 > 0 and every i ∈ N. From the Cauchy-Schwarz inequality, we get

3.37
Stationarity, association and our assumption 2.1 imply for some constant C 2 > 0 and every i ∈ N. Furthermore, i 1 ≤ j, thus, what follows from the inequalities

3.42
Now, let us estimate the last term in 3.35 : where
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

and by stationarity
Cov X 1 ; X ν .

3.46
Similarly, we deal with A 2 .We have

and by stationarity
Cov X 1 ; X ν i .

3.52
where Φ x is the standard normal distribution.By the inequality |Φ t − Φ t a | ≤ a/ 2π, we get

3.53
Combining the above considerations, we get for every ε > 0 lim sup 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, almost surely as k → ∞, consequently S k − kμ /k → 0 almost surely, as k → ∞.
n denotes the Euler constant.From 3.36 -3.40 , we get 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 By continuity of F and the same arguments as in 8 , we get the uniform convergence and the conclusion follows.