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

Authors: Przemysław Matuła; Iwona Stępień
Journal: ISRN Probability and Statistics
 (2012)

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## 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.


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## Body

## 1. Introduction

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=∑k=1nXk. 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(1.1)(∏i=1nSin!μn)1/γn⟶de2N,asn⟶∞,
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 have(1.2)lim⁡n→∞1log⁡⁡n∑k=1n1kI[(∏j=1kSjk!μk)1/(γk)≤x]=F(x),
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]. (Xn)n∈ℕ is a sequence of associated r.v., if for every finite subcollection Xi1,…,Xin and any coordinatewise nondecreasing functions f,g:ℝn→ℝ(1.3)Cov(f(Xi1,…,Xin),g(Xi1,…,Xin))≥0,
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(1.4)P(X≤x,Y≤y)≥P(X≤x)P(Y≤y),
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(Xn)n∈ℕ be a strictly stationary sequence of positive and square-integrable associated r.v. Assume that
(2.1)0<σ12:=σ2+2∑k=1∞Cov(X1,Xk+1)<∞.
Then,
(2.2)(∏k=1nSkn!μn)1/γ1n⟶de2N,asn⟶∞,
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 Theorem2.1. Our results generalize the ones obtained in [2, 3].Theorem 2.3.
Assume that the conditions of Theorem2.1 are satisfied, then,
(2.3)lim⁡n→∞sup⁡x≥0|1log⁡n∑i=1n1iI[(∏k=1iSki!μi)1/γ1i≤x]-F(x)|=0a.s.,
where (2.3)
F(x) denotes the distribution of the random variable (2.3)
exp⁡(2N).Remark 2.4.
Our Theorem2.3 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
(2.4)Cov(X1,Xn+1)=O(n-1(log⁡⁡n)-2-ε),forsomeε>0
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 Theorem2.1, 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 σn2 the variance of the sum ∑k=1n((Sk-kμ)/k), then,(3.1)σn2=Var⁡(∑k=1nbk,n(Xk-μ))=Var⁡(∑k=1nbk,nXk),
where bk,n=∑i=kn(1/i).The first lemma describes the asymptotic behavior of the varianceσn2.Lemma 3.1.
Assume that the conditions of Theorem2.1 are satisfied, then,
(3.2)lim⁡n→∞σn22n=σ12=σ2+2∑k=1∞Cov(X1,Xk+1).Proof.
Let us observe that, by stationarity, we have(3.3)σn22n=∑k=1nbk,n2Var⁡(Xk)2n+2∑k=1n-1fk,nCov(X1,Xk+1)2n,
where
(3.4)fk,n=b1,nb1+k,n+b2,nb2+k,n+⋯+bn-k,nbn,n=∑j=1n-kbj,nbj+k,n.
For k∈ℕ, let us put gk=∑i=k∞Cov(X1,Xi+1), then gk+1-gk=-Cov(X1,Xk+1). In order to calculate the second summand in (3.3), we shall apply the summation by parts formula
(3.5)∑k=1n-1fk,n(gk+1-gk)=(gnfn-1,n-f1,ng1)-∑k=1n-2gk+1(fk+1,n-fk,n).
Thus, in our case,(3.6)∑k=1n-1fk,nCov(X1,Xk+1)=f1,ng1-gnfn-1,n+∑k=1n-2gk+1(fk+1,n-fk,n).
Applying the identity
(3.7)∑j=1nbj,n2=2n-b1,n
(see formula (4) in [1]), we get
(3.8)f1,n=∑j=1n-1bj,nbj+1,n=∑j=1n-1bj,n(bj,n-1j)=∑j=1n-1bj,n2-∑j=1n-1bj,n1j=2n-b1,n-1n2-∑j=1n-1bj,n1j.
Thus, taking into account thatfn-1,n=b1,nbn,n, the formula (3.3) takes the following form:
(3.9)σn22n=σ2∑k=1nbk,n22n+2(f1,ng1-gnfn-1,n+∑k=1n-2gk+1(fk+1,n-fk,n))2n=σ2(2n-b1,n2n)+2n-b1,n-(1/n2)-∑j=1n-1bj,n(1/j)n∑k=1∞Cov(X1,Xk+1)-∑i=n∞Cov(X1,Xi+1)b1,nbn,nn-f1,n-fn-1,nn∑k=1n-2gk+1ck,n-2,
where ck,n-2, for k=1,2,…,n-2 are defined as follows:
(3.10)ck,n-2=fk,n-fk+1,n∑k=1n-2(fk,n-fk+1,n)=fk,n-fk+1,nf1,n-fn-1,n.
On account that fk,n>fk+1,n, we have ck,n-2>0. Furthermore, ∑k=1n-2ck,n-2=1. We shall prove that ck,n-2→0, as n→∞, for every k∈ℕ. At first, let us note that
(3.11)0<fk,n-fk+1,n=∑j=1n-kbj,nbj+k,n-∑j=1(n-k)-1bj,nbj+k+1,n=bn-k,nbn,n+∑j=1(n-k)-1bj,n1j+k=bn-k,nbn,n+∑j=1(n-k)-1∑i=jn1i1j+k<b1,nbn,n+∑j=1n∑i=1n1i1j=b1,nbn,n+(b1,n)2<2(1+log⁡⁡n)2,
by the inequality b1,n-logn<1. On the other hand, we have
(3.12)f1,n-fn-1,n=∑j=1n-1bj,nbj+1,n-b1,nbn,n=∑j=1n-1bj,n(bj,n-1j)-b1,nbn,n=∑j=1nbj,n2-bn,n2-∑j=1n-1bj,n1j-b1,nbn,n=2n-b1,n-1n2-∑j=1n-1bj,n1j-b1,n1n>2n-(2+log⁡⁡n)2>n,
for sufficiently large n (n≥29). Therefore, from (3.11) and (3.12), we get
(3.13)ck,n-2≤2(1+log⁡⁡n)2n⟶0,
for each k∈ℕ. From the Toeplitz theorem on the transformation of sequences into sequences (Problem  2.3.1 in [10]), we obtain
(3.14)∑k=1n-2gk+1ck,n-2⟶0,
since gk+1→0. It is easy to see that 0<f1,n-fn-1,n<∑j=1nbj,n2<2n.
Thus, we see that from (3.9) the conclusion follows.In the following lemma let us recall Theorem2.3from [11], which will be one of the tools needed in the sequel.Lemma 3.2.
Let{an,k;n∈ℕ,1≤k≤n} be a triangular array of real numbers satisfying
(3.15)sup⁡n∈N∑k=1nan,k2<∞,lim⁡n→∞max⁡1≤k≤n|an,k|=0.
Let (Yn)n∈ℕ be a sequence of centered, square-integrable associated r.v. such that the family {Yn2;n∈ℕ} is uniformly integrable and
(3.16)Var⁡(∑k=1nan,kYk)=1.
If furthermore∑j:|k-j|≥nCov(Yk,Yj)→0, uniformly in k∈ℕ, as n→∞, then ∑k=1nan,kYk→d𝒩, as n→∞.In the next lemma, we state the central limit theorem for sums of associated r.v. The proof particularly relies on Lemmas3.1 and 3.2.Lemma 3.3.
Assume that the conditions of Theorem2.1 are satisfied, then
(3.17)12nσ1∑k=1nSk-kμk=12nσ1∑k=1nbk,n(Xk-μ)⟶dN,asn⟶∞.Proof.
We apply Lemma3.2 to the sequence Yk=Xk-μ with an,k=bk,n/σn. Let us check the assumptions. Obviously the family {Yn2;n∈ℕ} is uniformly integrable. It is easy to see that by Lemma 3.1(3.18)∑k=1nan,k2=∑k=1nbk,n2σn2=2n-b1,n2n2nσn2⟶1σ12,asn⟶∞,
thus, supn∈ℕ∑k=1nan,k2<∞. Further we have max1≤k≤n|ank|=b1,n/σn≤(1+logn)/cn→0. Obviously Var(∑k=1nan,kYk)=1, thus,
(3.19)1σn∑k=1nbk,n(Xk-μ)⟶dN,12nσ1∑k=1nbk,n(Xk-μ)=12nσ1σn1σn∑k=1nbk,n(Xk-μ)⟶dN,
since we have σn2/2nσ12→1,asn→∞.We shall also need the following strong law of large numbers of the Marcinkiewicz-Zygmund type.Lemma 3.4.
Let(Xn)n∈ℕ be a strictly stationary sequence of square-integrable associated r.v. If  ∑j=2∞Cov(X1,Xj)<∞, then for every  p∈(0,2), one has
(3.20)1n1/p∑k=1n(Xk-EXk)⟶0,almostsurely,asn⟶∞.Proof.
We directly apply Theorem  3.3 in [12]. This result states that if (bn)n∈ℕ is a nondecreasing sequence of positive real numbers and (Xn)n∈ℕ is a sequence of associated r.v. with
(3.21)∑j=1∞Var⁡(Xj)bj2+∑j≠k∞Cov(Xj,Xk)bjbk<∞,
then, bn-1∑j=1n(Xj-EXj)→0 almost surely, as n→∞.Proof of Theorem2.1.
After taking the logarithm it suffices to show that(3.22)L:=1γ12n∑k=1nlog⁡(Skkμ)⟶dN.
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
(3.23)L=1γ12n∑k=1nSk-kμkμ-1γ12n∑k=1n((Sk-kμ)/kμ)22(1+θk((Sk-kμ)/kμ))2,
where θk 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 (Sk-kμ)/k3/4)→0,almostsurelyask→∞, consequently (Sk-kμ)/k→0 almost surely, as k→∞. Therefore,
(3.24)((Sk-kμ)/k3/4)2(1+θk((Sk-kμ)/kμ))2⟶0,almostsurely,ask→∞.
The weighted average with weights 1/k is also convergent to 0, accordingly
(3.25)1γ12n∑k=1n((Sk-kμ)/kμ)2(1+θk((Sk-kμ)/kμ))2=1γ12n∑k=1n1k((Sk-kμ)/k3/4μ)2(1+θk((Sk-kμ)/kμ))2⟶0,
almost surely, as n→∞, and the proof is completed.In the proof of Theorem2.3, we shall use the following inequality concerning indicators. The proof is immediate.Lemma 3.5.
For any random variablesX,  Y and real numbers  t∈ℝ,  a>0, the following inequalities hold:
(3.26)I[X+Y≤t]≤I[X≤t+a]+I[|Y|>a],I[X+Y≤t]≥I[X≤t-a]-I[|Y|>a].
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(an)n∈ℕ be a sequence of positive numbers, one puts  bn=∑k=1nak, and assume that  an/bn→0,bn→∞ as  n→∞. Let  (Yn)n∈ℕ be a sequence of pairwise PQD r.v. with distribution functions  Fn(x), respectively. Suppose that  Fn→dF, where  F is a continuous distribution function, and
(3.27)∑j=1∞∑i=1jaiajCov(Yi,Yj)bj2<∞.
Then,
(3.28)sup⁡-∞<x<∞|1bn∑i=1naiI[Yi≤x]-F(x)|⟶0a.s.,asn⟶∞.Proof of Theorem2.3.
Letε>0 be fixed. We put Ck=Sk/kμ and logx=t. By using the expansion of the logarithm and Lemma 3.5, we get
(3.29)I[(∏k=1iSki!μi)1/γ1i≤x]=I[1γ1i∑k=1i(Ck-1)+Ri≤t]≤I[1γ1i∑k=1i(Ck-1)≤t+ε]+I[|Ri|>ε],I[(∏k=1iSki!μi)1/γ1i≤x]≥I[1γ1i∑k=1i(Ck-1)≤t-ε]-I[|Ri|>ε],
where Ri=(-1/γ1i)∑k=1i((Ck-1)2/2(1+θk(Ck-1))2). As in the proof of Theorem 2.1, we see that Ri→0 almost surely, as i→∞. Thus, for almost every ω∈Ω there exists i0(ω) such that for i≥i0 we have |Ri(ω)|<ε, therefore,
(3.30)I[|Ri(ω)|>ε]⟶0,i⟶∞,
for almost every ω. Now, we shall apply Lemma 3.6 to the sequence
(3.31)Yi=1γ1i∑k=1i(Ck-1),
and weights ai=1/i and bn=logn. The r.v. Yi are not only PQD but associated as well, as sums of the r.v. Ck-1 which are associated. By Lemma 3.3, we have
(3.32)1γ1i∑k=1i(Ck-1)⟶d2N,asi⟶∞.
We shall check the condition (3.27). Let us observe that
(3.33)Yi=1γ1i∑k=1i(Ck-1)=1σ1i∑k=1ibk,i(Xk-μ),
whereas before bk,i=∑ν=ki(1/ν).
For1≤i<j, we have the following decomposition:
(3.34)∑k=1jbk,j(Xk-μ)=∑k=1i(bk,i+bi+1,j)(Xk-μ)+∑k=i+1jbk,j(Xk-μ)=∑k=1ibk,i(Xk-μ)+bi+1,j∑k=1i(Xk-μ)+∑k=i+1jbk,j(Xk-μ).
Thus, for 1≤i<j, we get
(3.35)0≤Cov(Yi,Yj)=1σ12ij(Var⁡(∑k=1ibk,i(Xk-μ))+Cov(∑k=1ibk,i(Xk-μ);bi+1,j∑k=1i(Xk-μ))+Cov(∑k=1ibk,i(Xk-μ);∑k=i+1jbk,j(Xk-μ))).
We shall find the bounds for each of the above terms. By Lemma 3.1, we have
(3.36)Var⁡(∑k=1ibk,i(Xk-μ))=σi2≤C1i,
for some constant C1>0 and every i∈ℕ. From the Cauchy-Schwarz inequality, we get
(3.37)Cov(∑k=1ibk,i(Xk-μ);bi+1,j∑k=1i(Xk-μ))≤Var⁡(∑k=1ibk,i(Xk-μ))⋅bi+1,jVar⁡(∑k=1i(Xk-μ)).
Stationarity, association and our assumption (2.1) imply
(3.38)lim⁡i→∞1iVar⁡(∑k=1i(Xk-μ))=σ12,
thus,
(3.39)Var⁡(∑k=1i(Xk-μ))≤C2i,
for some constant C2>0 and every i∈ℕ. Furthermore, i+1≤j, thus,
(3.40)bi+1,j=∑ν=1j1ν-∑ν=1i1ν≤1+log⁡⁡j-γE-log⁡⁡i≤1+log⁡⁡ji≤p(ji)1/p,
what follows from the inequalities
(3.41)γE≤∑ν=1n1ν-log⁡⁡n≤1,1+log⁡⁡x<px1/p,p>1,x≥1,
where γE=limn→∞(∑k=1n(1/k)-logn) denotes the Euler constant. From (3.36)–(3.40), we get
(3.42)Cov(∑k=1ibk,i(Xk-μ);bi+1,j∑k=1i(Xk-μ))≤C1C2⋅p⋅i⋅(ji)1/p.
Now, let us estimate the last term in (3.35):
(3.43)A:=Cov(∑k=1ibk,i(Xk-μ);∑k=i+1jbk,j(Xk-μ))=A1+A2,say,
where
(3.44)A1=∑ν=2i∑s=1min⁡(i-1,j-i)Cov(Xν+s-1;Xi+s)bν+s-1,ibi+s,j,A2=∑ν=1j-i∑s=1min⁡(i,j-i)Cov(Xs;Xν+s+i-1)bs,ibν+s+i-1,j,
for i=1, we put A1=0.
By the monotonicity properties of the coefficientsbn,m with respect to n and fixed m, Cauchy-Schwarz inequality, (3.40), and formula (3.7), we get
(3.45)∑s=1min⁡(i-1,j-i)bν+s-1,ibi+s,j≤∑s=1min⁡(i-1,j-i)bs,ibi+s,j≤∑s=1min⁡(i-1,j-i)bs,i2⋅∑s=1min⁡(i-1,j-i)bi+s,j2≤∑s=1ibs,i2⋅∑s=1min⁡(i-1,j-i)bi+1,j2≤p2⋅i⋅(ji)1/p,
and by stationarity
(3.46)A1≤p2⋅i⋅(ji)1/p∑ν=2iCov(Xi+1;Xν)=p2⋅i⋅(ji)1/p∑ν=2iCov(X1;Xν).
Similarly, we deal with A2. We have
(3.47)∑s=1min⁡(i,j-i)bs,ibν+s+i-1,j≤∑s=1min⁡(i,j-i)bs,i2⋅∑s=1min⁡(i,j-i)bν+s+i-1,j2≤∑s=1ibs,i2⋅∑s=1ibi+1,j2≤p2⋅i⋅(ji)1/p,
and by stationarity
(3.48)A2≤p2⋅i⋅(ji)1/p∑ν=1j-iCov(X1;Xν+i).
Therefore, from (3.46), (3.48), and (2.1), we get
(3.49)A≤p2⋅i⋅(ji)1/p∑ν=2jCov(X1;Xν)≤σ12p2⋅i⋅(ji)1/p.
Clearly, Cov(Yi,Yi) is bounded by some constant for all i≥1. So that from (3.35), (3.36), (3.42) and (3.49), we obtain
(3.50)Cov(Yi,Yj)≤C3(ij)1/2-1/p,
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
(3.51)∑j=1∞∑i=1j1ijlog⁡2jCov(Yi,Yj)≤C3∑j=1∞1jlog⁡2j1j(1/2)-(1/p)∑i=1j1ii(1/2)-(1/p)≤C4∑j=1∞1jlog⁡2j1j(1/2)-(1/p)j(1/2)-(1/p)<∞,
for some C4>0, since ∑i=1ji-(1/2)-(1/p) is of order j(1/2)-(1/p). By (3.28), we get
(3.52)1log⁡n∑i=1n1iI[Yi≤t+ε]⟶Φ(t+ε2)a.s.,asn⟶∞,1log⁡n∑i=1n1iI[Yi≤t-ε]⟶Φ(t-ε2)a.s.,asn⟶∞,
where Φ(x) is the standard normal distribution. By the inequality |Φ(t)-Φ(t+a)|≤a/2π, we get
(3.53)Φ(t+ε2)<Φ(t2)+ε,Φ(t-ε2)>Φ(t2)-ε.
Combining the above considerations, we get for every ε>0(3.54)limsup⁡n→∞1log⁡⁡n∑i=1n1iI[(∏k=1iSki!μi)1/γ1i≤x]≤Φ(t2)+εa.s.,liminf⁡n→∞1log⁡⁡n∑i=1n1iI[(∏k=1iSki!μi)1/γ1i≤x]≥Φ(t2)-εa.s.,
since ε>0 was arbitrary, we conclude that
(3.55)lim⁡n→∞1log⁡⁡n∑i=1n1iI[(∏k=1iSki!μi)1/γ1i≤x]=Φ(log⁡⁡x2)a.s.
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.



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