# Strong Laws of Large Numbers for Arrays of Rowwise NA and LNQD Random Variables

Authors: Jiangfeng Wang; Qunying Wu
Journal: Journal of Probability and Statistics
 (2011)

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## Abstract

Some strong laws of large numbers and strong convergence properties for arrays of rowwise negatively associated and linearly negative quadrant dependent random variables are obtained. The results obtained not only generalize the result of Hu and Taylor to negatively associated and linearly negative quadrant dependent random variables, but also improve it.


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## Body

## 1. Introduction

Let{Xn}n∈ℕ be a sequence of independent distributed random variables. The Marcinkiewicz-Zygmund strong law of large numbers (SLLN) provides that(1.1)1n1/α∑i=1n(Xi-EXi)⟶0a.s.for1≤α<2,1n1/α∑i=1nXi⟶0a.s.for0<α<1asn⟶∞
if and only if E|X|α<∞. The case α=1 is due to Kolmogorov. In the case of independence (but not necessarily identically distributed), Hu and Taylor [1] proved the following strong law of large numbers.Theorem 1.1.
Let{Xni,1≤i≤n,n≥1} be a triangular array of rowwise independent random variables. Let {an}n∈ℕ be a sequence of positive real numbers such that 0<an↑∞. Let ψ(t) be a positive, even function such that ψ(t)/|t|p is an increasing function of |t| and ψ(t)/|t|p+1 is a decreasing function of |t|, respectively, that is,
(1.2)ψ(t)|t|p↑,ψ(t)|t|p+1↓,as|t|↑,
for some positive integer p. If p≥2 and
(1.3)EXni=0,∑n=1∞∑i=1nEψ(|Xni|)ψ(|an|)<∞,∑n=1∞(∑i=1nE(Xnian)2)2k<∞,
where k is a positive integer, then
(1.4)1an∑i=1nXni⟶0a.s.Definition 1.2 (cf. [2]).
A finite family of random variables{Xn}n∈ℕ is said to be negatively associated (NA, in short) if, for any disjoint subsets A and B of {1,2,…,n} and any real coordinate-wise nondecreasing functions f on on ℝA and g on ℝB,
(1.5)Cov(f(Xi,i∈A),g(Yj,j∈B))≤0,
whenever the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA. This concept was introduced by Joag-Dev and Proschan [2].Definition 1.3 (cf. [3, 4]).
Two random variablesX and Y are said to be negative quadrant dependent (NQD, in short) if, for any x,y∈ℝ,
(1.6)P(X<x,Y<y)≤P(X<x)P(Y<y).
A sequence {Xn}n∈ℕ of random variables is said to be pairwise NQD if Xi and Xj are NQD for all i,j∈ℕ+ and i≠j.Definition 1.4 (cf. [5]).
A sequence{Xn}n∈ℕ of random variables is said to be linearly negative quadrant dependent (LNQD, in short) if, for any disjoint subsets A,B∈ℤ+ and positive rj′s,
(1.7)∑k∈ArkXk,∑j∈BrjXjareNQD.Remark 1.5.
It is easily seen that if{Xn}n∈ℕ is a sequence of LNQD random variables, then {aXn+b}n∈ℕ is still a sequence of LNQD random variables, where a and b are real numbers.The NA property has aroused wide interest because of numerous applications in reliability theory, percolation theory, and multivariate statistical analysis. In the past decades, a lot of effort was dedicated to proving the limit theorems of NA random variables. A Kolmogorov-type strong law of large numbers of NA random variables was established by Matuła in [6], which is the same as I.I.D. sequence, and Marcinkiewicz-type strong law of large Numbers was obtained by Su and Wang [7] for NA random variable sequence with assumptions of identical distribution; Yang et al. [8] gave the strong law of large Numbers of a general method.The concept of LNQD sequence was introduced by Newman [5]. Some applications for LNQD sequence have been found. See, for example, Newman [5] who established the central limit theorem for a strictly stationary LNQD process. Wang and Zhang [9] provided uniform rates of convergence in the central limit theorem for LNQD sequence. Ko et al. [10] obtained the Hoeffding-type inequality for LNQD sequence. Ko et al. [11] studied the strong convergence for weighted sums of LNQD arrays, and so forth.The aim of this paper is to establish a strong law of large numbers for arrays of NA and LNQD random variables. The result obtained not only extends Theorem1.1 for independent sequence above to the case of NA and LNQD random variables sequence, but also improves it.Lemma 1.6 (cf. [12]).
Let{Xn,n≥1} be NA random variables, EXn=0, E|Xn|q<∞, n≥1, q≥2. Then, there exists a positive constant c such that
(1.8)Emax⁡1≤i≤n|∑i=1kXi|q≤c[∑i=1nE|Xi|q+(∑i=1nEXi2)q/2],∀n≥1.Letc denote a positive constant which is not necessary the same in its each appearance.Lemma 1.7 (cf. [3, 4]).
Let random variablesX and Y be NQD, then
(1.9)(1)EXY≤EXEY;(2)P(X<x,Y<y)≤P(X<x)P(Y<y);(3)iffandgarebothnondecreasing(orboth non increasing)functions,thenf(X)andg(Y)areNQD.Lemma 1.8.
Let{Xn,n≥1} be LNQD random variables sequences with mean zero and 0<Bn=∑k=1nEXk2. Then,
(1.10)P(|Sn|≥x)≤∑k=1nP(|Sk|≥y)+2exp⁡⁡(xy-xylog⁡⁡(1+xyBn)),
for any x>0, y>0.This lemma is easily proved by following Fuk and Nagaev [13]. Here, we omit the details of the proof.


## 2. Main Results

Theorem 2.1.
Let{Xni;i≥1,n≥1} be an array of rowwise NA random variables. Let {an}n∈ℕ be a sequence of positive real numbers such that 0<an↑∞. Let ψ(t) be a positive, even function such that ψ(t)/|t| is an increasing function of |t| and ψ(t)/|t|p is a decreasing function of |t|, respectively, that is,
(2.1)ψ(t)|t|↑,ψ(t)|t|p↓,as|t|↑
for some nonnegative integer P. If p≥2 and
(2.2)EXni=0,∑n=1∞∑i=1nEψ(|Xni|)ψ(|an|)<∞,∑n=1∞(∑i=1nE(Xnian)2)v/2<∞,
where v is a positive integer and v≥p, then
(2.3)∑n=1∞P(max⁡1≤k≤n|1an∑i=1kXni|>ɛ)<∞,foranyɛ>0.Proof of Theorem2.1.
For alli≥1, letXi(n)=-anI(Xni<-an)+XniI(|Xni|≤an)+anI(Xni>an), Tj(n)=(1/an)∑i=1j(Xi(n)-EXi(n)), then, for all ɛ>0,
(2.4)P(max⁡1≤k≤n|1an∑i=1kXni|>ɛ)≤P(max⁡1≤j≤n|Xnj|>an)+P(max⁡1≤j≤n|Tj(n)|>ɛ-max⁡1≤j≤n|1an∑i=1jEXi(n)|).
First, we show that
(2.5)max⁡1≤j≤n|1an∑i=1jEXi(n)|⟶0,asn⟶∞.
In fact, by EXni=0, ψ(t)/|t|↑ as |t|↑ and Σn=1∞Σi=1nE(ψ(|Xni|)/ψ(an))<∞, then
(2.6)max⁡1≤j≤n|1an∑i=1jEXi(n)|≤max⁡1≤j≤n1an(|∑i=1jEXniI(|Xni|≤an)|+|∑i=1jE(anI(|Xni|>an))|)≤max⁡1≤j≤n1an(∑i=1j|EXniI(|Xni|≤an)|+|∑i=1jE(anI(|Xni|>an))|)=max⁡1≤j≤n1an(∑i=1j|EXniI(|Xni|>an)|+|∑i=1jE(anI(|Xni|>an))|)≤2∑i=1nE|Xni|I(|Xni|>an)an≤2∑i=1nEψ(|Xni|)I(|Xni|>an)ψ(an)≤2∑i=1nEψ(|Xni|)ψ(an)⟶0,asn⟶∞.
From (2.4) and (2.5), it follows that, for n sufficiently large,
(2.7)P(max⁡1≤k≤n|1an∑i=1kXni|>ɛ)≤∑j=1nP(|Xnj|>an)+P(max⁡1≤j≤n|Tj(n)|>ɛ2).
Hence, we need only to prove that
(2.8)I≕∑n=1∞∑j=1nP(|Xnj|>an)<∞,II≕∑n=1∞P(max⁡1≤j≤n|Tj(n)|>ɛ2)<∞.
From the fact that Σn=1∞Σi=1nE(ψ(|Xni|)/ψ(an))<∞, it follows easily that
(2.9)I=∑n=1∞∑j=1nP(|Xnj|>an)≤∑n=1∞∑j=1nEψ(|Xnj|)ψ(an)<∞.
By v≥p and ψ(t)/|t|p↓ as |t|↑, then ψ(t)/|t|v↓ as |t|↑.
By the Markov inequality, Lemma1.6, and ∑n=1∞(∑i=1nE(Xni/an)2)v/2<∞, we have
(2.10)II=∑n=1∞P(max⁡1≤j≤n|Tj(n)|>ɛ2)≤∑n=1∞(ɛ2)-vEmax⁡1≤j≤n|Tj(n)|v≤c(ɛ2)-v∑n=1∞1anv[(∑j=1nE|Xj(n)|2)v/2+∑j=1nE|Xj(n)|v]≤c∑n=1∞1anv∑j=1nE|Xj(n)|v+c∑n=1∞1anv(∑j=1nE|Xj(n)|2)v/2≤c(∑n=1∞1anv∑j=1nE|Xnj|vI(|Xnj|≤an)+I)+c∑n=1∞1anv(∑j=1nE|Xj(n)|2)v/2≤c∑n=1∞∑i=1nEψ(|Xni|)ψ(an)+c∑n=1∞1anv(∑j=1nE|Xj(n)|2)v/2≤c∑n=1∞∑i=1nEψ(|Xni|)ψ(an)+c∑n=1∞(∑j=1nE(Xnian)2)v/2<∞.
Now we complete the proof of Theorem 2.1.Corollary 2.2.
Under the conditions of Theorem2.1, then
(2.11)1an∑i=1nXni⟶0a.s.Proof of Corollary2.2.
By Theorem2.1, the proof of Corollary 2.2 is obvious.Remark 2.3.
Corollary2.2 not only generalizes the result of Hu and Taylor to NA random variables, but also improves it.Theorem 2.4.
Let{Xni;i≥1,n≥1} be an array of rowwise LNQD random variables. Let {an}n∈ℕ be a sequence of positive real numbers such that 0<an↑∞. Let ψ(t) be a positive, even function such that ψ(t)/|t| is an increasing function of |t| and ψ(t)/|t|p is a decreasing function of |t|, respectively, that is,
(2.12)ψ(t)|t|↑,ψ(t)|t|p↓,as|t|↑
for some positive integer p. If 1<p≤2 and
(2.13)EXni=0,∑n=1∞∑i=1nEψ(|Xni|)ψ(an)<∞,
then
(2.14)∑n=1∞P(|1an∑i=1nXni|>ɛ)<∞,foranyɛ>0.Proof of Theorem2.4.
For any1≤k≤n, n≥1, let
(2.15)Ynk=-anI(Xnk<-an)+XnkI(|Xnk|≤an)+anI(Xnk>an),Znk=Xnk-Ynk=(Xnk+an)I(Xnk<-an)+(Xnk-an)I(Xnk>an).
To prove (2.14), it suffices to show that
(2.16)1an∑k=1nZnk⟶0completely,(2.17)1an∑k=1n(Ynk-EYnk)⟶0completely,(2.18)1an∑k=1nEYnk⟶0asn⟶∞.
Firstly, we prove (2.16):
(2.19)∑n=1∞P(1an|∑k=1nZnk|>ɛ)≤∑n=1∞E|∑k=1nZnk|anɛ≤C∑n=1∞∑k=1nE|Xnk|I(|Xnk|>an)an≤C∑n=1∞∑i=1nEψ(|Xni|)ψ(an)<∞.
Secondly, we prove (2.17). By Lemma 1.7, we know that {Ynk-EYnk,1≤k≤n,n≥1} is an array of rowwise LNQD mean zero random variables. Let Bn′=∑k=1nE(Ynk-EYnk)2. Take x=ɛan, y=2ɛan/v, and v≥1. By Lemma 1.8, for all ɛ>0,
(2.20)∑n=1∞P(1an|∑k=1n(Ynk-EYnk)|>ɛ)≤∑n=1∞∑k=1nP(|Ynk-EYnk|>ɛanη)+C∑n=1∞(Bn′Bn′+ɛ2an2/η)η∶=I1+I2.
From (2.12), (2.13), the Markov inequality, and Cr-inequality,
(2.21)I1=∑n=1∞∑k=1nP(|Ynk-EYnk|>ɛanη)≤C∑n=1∞∑k=1nE|Ynk-EYnk|panp≤C∑n=1∞∑k=1nE|Ynk|panp≤C∑n=1∞∑k=1nEψk(|Ynk|)ψk(an)≤C∑n=1∞∑k=1nEψk(|Xnk|)ψk(an)<∞.
Note that |Ynk|≤|Xnk|, η≥1 and 1<p≤2. From (2.12), (2.13), and the Cr-inequality,
(2.22)I2≤C∑n=1∞(∑k=1nan-2E(Ynk-EYnk)2)η≤C∑n=1∞(∑k=1nE|Ynk|panp)η≤C(∑n=1∞∑k=1nE|Ynk|panp)η≤C(∑n=1∞∑k=1nEψk(|Ynk|)ψk(an))η≤C(∑n=1∞∑k=1nEψk(|Xnk|)ψk(an))η<∞.
Finally, we prove (2.18). For 1≤k≤n, n≥1, EXnk=0, then EYnk=-EZnk. From the definition of Znk if Xnk>an, then 0<Znk=Xnk-an<Xnk, if Xnk<-an, then Xnk<Znk=Xnk+an<0. So |Znk|≤|Xnk|I(|Xnk|>an). Consequently,
(2.23)1an|∑k=1nEYnk|=1an|∑k=1nEZnk|≤∑k=1nE|Znk|an≤∑k=1nE|Xnk|I(|Xnk|>an)an≤∑k=1nEψk(Xnk)ψk(an)⟶0asn⟶∞.
The proof is completed.Theorem 2.5.
Let{Xni;i≥1,n≥1} be an array of rowwise LNQD random variables. Let {an}n∈ℕ, be a sequence of positive real numbers such that 0<an↑∞. Let {ψn(t)}n∈ℕ, be a sequence of positive even functions and satisfy (2.12) for p>2. Suppose that
(2.24)∑n=1∞(∑i=1nE(Xnian)2)v/2<∞,
where v is a positive integer, v≥p, the conditions (2.13) and (2.24) imply (2.24).Proof of Theorem2.5.
Following the notations and the methods of the proof in Theorem2.4, (2.16), (2.18), and I1<∞ hold. So, we only need to show that I2<∞. Let η>v/2. By (2.24), we have
(2.25)I2≤∑n=1∞(∑k=1nan-2E(Ynk-EYnk)2)η≤C∑n=1∞[(∑k=1nEYnk2an2)v/2]2η/v≤C[∑n=1∞(∑k=1nEXnk2an2)v/2]2η/v<∞.
The proof is completed.Corollary 2.6.
Under the conditions of Theorem2.4 or Theorem 2.5, then
(2.26)1an∑i=1nXni⟶0a.s.Remark 2.7.
Because of the maximal inequality of LNQD, the result of LNQD we have obtained generalizes and improves the result of Hu and Taylor.



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*Source: 708087-2011-12-01.xml*