# A Note on the Strong Law of Large Numbers for Arrays of Rowwiseρ˜-Mixing Random Variables

Authors: Aiting Shen; Shuhe Hu
Journal: Discrete Dynamics in Nature and Society
 (2011)

---

## Abstract

Let{Xni,i≥1,n≥1} be an array of rowwise ρ˜-mixing random
variables. Some strong law of large numbers for arrays of rowwise ρ˜-mixing
random variables is studied under some simple and weak conditions.


---

## Body

## 1. Introduction

Let{X,Xn,n≥1} be a sequence of independent and identically distributed random variables. The Marcinkiewicz-Zygmund strong law of large numbers states that (1.1)1nα∑i=1n(Xi-EXi)⟶0a.s.for1≤α<2,1nα∑i=1nXi⟶0a.s.for0<α<1
if and only if E|X|α<∞. In the case of independence, 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≥1} be a sequence of positive real numbers such that 0<an↑∞. Let g(t) be a positive, even function such that g(|t|)/|t|p is an increasing function of |t| and g(|t|)/|t|p+1 is a decreasing function of |t|, respectively, that is,
(1.2)g(|t|)|t|p↑,g(|t|)|t|p+1↓,as|t|↑
for some nonnegative integer p. If p≥2 and
(1.3)EXni=0,∑n=1∞∑i=1nEg(|Xni|)g(an)<∞,∑n=1∞(∑i=1nE(Xnian)2)2k<∞,
where k is a positive integer, then
(1.4)1an∑i=1nXni⟶0a.s.asn⟶∞.Zhu [2] generalized and improved the result of Hu and Taylor [1] for triangular arrays of rowwise independent random variables to the case of arrays of rowwise ρ̃-mixing random variables as follows.Theorem 1.2.
Let{Xni:i≥1,n≥1} be an array of rowwise ρ̃-mixing random variables. Let {an,n≥1} be a sequence of positive real numbers such that 0<an↑∞. Let g(t) be a positive, even function such that g(|t|)/|t| is an increasing function of |t| and g(|t|)/|t|p is a decreasing function of |t|, respectively, that is,
(1.5)g(|t|)|t|↑,g(|t|)|t|p↓,as|t|↑
for some nonnegative integer p. If p≥2 and
(1.6)EXni=0,∑n=1∞∑i=1nEg(|Xni|)g(an)<∞,∑n=1∞(∑i=1nE(Xnian)2)v/2<∞,
where v is a positive integer, v≥p, then
(1.7)∀ε>0,∑n=1∞P(max⁡1≤j≤n|1an∑i=1jXni|>ε)<∞.In the following, we will give the definitions of aρ̃-mixing sequence and the array of rowwise ρ̃-mixing random variables.Let{Xn,n≥1} be a sequence of random variables defined on a fixed probability space (Ω,ℱ,P). Write ℱ𝒮=σ(Xi,i∈S⊂ℕ). For any given σ-algebras ℬ, ℛ in ℱ, let (1.8)ρ(B,R)=sup⁡X∈L2(B),Y∈L2(R)|EXY-EXEY|(Var⁡XVar⁡Y)1/2.
Define the ρ̃-mixing coefficients by (1.9)ρ̃(k)=sup⁡{ρ(FS,FT):S,TarefinitesubsetsofNsuchthatdist⁡(S,T)≥k},k≥0.
Obviously, 0≤ρ̃(k+1)≤ρ̃(k)≤1 and ρ̃(0)=1.Definition 1.3.
A sequence{Xn,n≥1} of random variables is said to be a ρ̃-mixing sequence if, there exists k∈ℕ such that ρ̃(k)<1.An array of random variables{Xni,i≥1,n≥1} is called rowwise ρ̃-mixing if for every n≥1, {Xni,i≥1} is a ρ̃-mixing sequence of random variables.Theρ̃-mixing random variables were introduced by Bradley [3], and many applications have been found. ρ̃-mixing is similar to ρ-mixing, but both are quite different. Many authors have studied this concept providing interesting results and applications. See, for example, Zhu [2], An and Yuan [4], Kuczmaszewska [5], Bryc and Smoleński [6], Cai [7], Gan [8], Peligrad [9, 10], Peligrad and Gut [11], Sung [12], Utev and Peligrad [13], Wu and Jiang [14], and so on. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.The main purpose of this paper is to further study the strong law of large numbers for arrays of rowwiseρ̃-mixing random variables. We will introduce some simple conditions to prove the strong law of large numbers. The techniques used in the paper are inspired by Zhu [2].


## 2. Main Results

Throughout the paper, letI(A) be the indicator function of the set A. C denotes a positive constant which may be different in various places.The proofs of the main results of this paper are based upon the following lemma.Lemma 2.1 (Utev and Peligrad [13, Theorem 2.1]).
Let{Xn,n≥1} be a ρ̃-mixing sequence of random variables, EXi=0, E|Xi|p<∞ for some p≥2 and for every i≥1. Then, there exists a positive constant C depending only on p such that
(2.1)E(max⁡1≤j≤n|∑i=1jXi|p)≤C{∑i=1nE|Xi|p+(∑i=1nEXi2)p/2}.As for arrays of rowwiseρ̃-mixing random variables {Xni:i≥1,n≥1}, we assume that the constant C from Lemma 2.1 is the same for each row throughout the paper. Our main results are as follows.Theorem 2.2.
Let{Xni:i≥1,n≥1} be an array of rowwise ρ̃-mixing random variables and let {an,n≥1} be a sequence of positive real numbers. Let {gn(t),n≥1} be a sequence of positive, even functions such that gn(|t|) is an increasing function of |t| and gn(|t|)/|t| is a decreasing function of |t| for every n≥1, respectively, that is,
(2.2)gn(|t|)↑,gn(|t|)|t|↓,as|t|↑.
If
(2.3)∑n=1∞∑i=1nEgn(|Xni|)gn(an)<∞,
then, for any ε>0,
(2.4)∑n=1∞P(max⁡1≤j≤n|1an∑i=1jXni|>ε)<∞.Proof.
For fixedn≥1, define
(2.5)Xi(n)=XniI(|Xni|≤an),i≥1,Tj(n)=1an∑i=1j(Xi(n)-EXi(n)),j=1,2,…,n.
It is easy to check that for any ε>0,
(2.6)(max⁡1≤j≤n|1an∑i=1jXni|>ε)⊂(max⁡1≤i≤n|Xni|>an)⋃(max⁡1≤j≤n|1an∑i=1jXi(n)|>ε),
which implies that
(2.7)P(max⁡1≤j≤n|1an∑i=1jXni|>ε)≤P(max⁡1≤i≤n|Xni|>an)+P(max⁡1≤j≤n|1an∑i=1jXi(n)|>ε)≤∑i=1nP(|Xni|>an)+P(max⁡1≤j≤n|Tj(n)|>ε-max⁡1≤j≤n|1an∑i=1jEXi(n)|).
Firstly, we will show that
(2.8)max⁡1≤j≤n|1an∑i=1jEXi(n)|⟶0,asn⟶∞.
Actually, by conditions gn(|t|)/|t|↓ as |t|↑ and (2.3), we have that
(2.9)max⁡1≤j≤n|1an∑i=1jEXi(n)|=max⁡1≤j≤n|1an∑i=1jEXniI(|Xni|≤an)|≤1an∑i=1nE|Xni|I(|Xni|≤an)≤∑i=1nEgn(|Xni|)I(|Xni|≤an)gn(an)≤∑i=1nEgn(|Xni|)gn(an)⟶0,asn⟶∞,
which implies (2.8). It follows from (2.7) and (2.8) that for n large enough,
(2.10)P(max⁡1≤j≤n|1an∑i=1jXni|>ε)≤∑i=1nP(|Xni|>an)+P(max⁡1≤j≤n|Tj(n)|>ε2).
Hence, to prove (2.4), we only need to show that
(2.11)∑n=1∞∑i=1nP(|Xni|>an)<∞,(2.12)∑n=1∞P(max⁡1≤j≤n|Tj(n)|>ε2)<∞.
The conditions gn(|t|)↑ as |t|↑ and (2.3) yield that
(2.13)∑n=1∞∑i=1nP(|Xni|>an)≤∑n=1∞∑i=1nEgn(|Xni|)gn(an)<∞,
which implies (2.11).
By Markov's inequality, Lemma2.1 (for p=2), Cr's inequality, gn(|t|)/|t|↓ as |t|↑ and (2.3), we can get that(2.14)∑n=1∞P(max⁡1≤j≤n|Tj(n)|>ε2)≤C∑n=1∞E(max⁡1≤j≤n|Tj(n)|2)≤C∑n=1∞1an2∑i=1nE|Xi(n)-EXi(n)|2≤C∑n=1∞1an2∑i=1nE|Xi(n)|2=C∑n=1∞∑i=1nE|Xni|2I(|Xni|≤an)an2≤C∑n=1∞∑i=1nE|Xni|I(|Xni|≤an)an≤C∑n=1∞∑i=1nEgn(|Xni|)I(|Xni|≤an)gn(an)≤C∑n=1∞∑i=1nEgn(|Xni|)gn(an)<∞,
which implies (2.12). This completes the proof of the theorem.Corollary 2.3.
Under the conditions of Theorem2.2,
(2.15)1an∑i=1nXni⟶0a.s.asn⟶∞.Theorem 2.4.
Let{Xni:i≥1,n≥1} be an array of rowwise ρ̃-mixing random variables and let {an,n≥1} be a sequence of positive real numbers. Let {gn(t),n≥1} be a sequence of nonnegative, even functions such that gn(|t|) is an increasing function of |t| for every n≥1. Assume that there exists a constant δ>0 such that gn(t)≥δt for 0<t≤1. If
(2.16)∑n=1∞∑i=1nEgn(Xnian)<∞,
then for any ε>0, (2.4) holds true.Proof.
We use the same notations as that in Theorem2.2. The proof is similar to that of Theorem 2.2.
Firstly, we will show that (2.8) holds true. In fact, by the conditions gn(t)≥δt for 0<t≤1 and (2.16), we have that(2.17)max⁡1≤j≤n|1an∑i=1jEXi(n)|=max⁡1≤j≤n|1an∑i=1jEXniI(|Xni|≤an)|≤∑i=1nE(|Xni|anI(|Xni|≤an))≤1δ∑i=1nEgn(Xnian)I(|Xni|≤an)≤1δ∑i=1nEgn(Xnian)⟶0,asn⟶∞,
which implies (2.8).
According to the proof of Theorem2.2, we only need to prove that (2.11) and (2.12) hold true.
When|Xni|>an>0, we have gn(Xni/an)≥gn(1)≥δ, which yields that(2.18)P(|Xni|>an)=EI(|Xni|>an)≤1δEgn(Xnian).
Hence,
(2.19)∑n=1∞∑i=1nP(|Xni|>an)≤1δ∑n=1∞∑i=1nEgn(Xnian)<∞,
which implies (2.11).
By Markov's inequality, Lemma2.1 (for p=2), Cr's inequality, gn(t)≥δt for 0<t≤1 and (2.16), we can get that(2.20)∑n=1∞P(max⁡1≤j≤n|Tj(n)|>ε2)≤C∑n=1∞∑i=1nE|Xni|2I(|Xni|≤an)an2≤C∑n=1∞∑i=1nE|Xni|I(|Xni|≤an)an≤C∑n=1∞∑i=1nEgn(Xnian)I(|Xni|≤an)≤C∑n=1∞∑i=1nEgn(Xnian)<∞,
which implies (2.12). This completes the proof of the theorem.Corollary 2.5.
Under the conditions of Theorem2.4,
(2.21)1an∑i=1nXni⟶0a.s.asn⟶∞.Corollary 2.6.
Let{Xni,i≥1,n≥1} be an array of rowwise ρ̃-mixing random variables and let {an,n≥1} be a positive real numbers. If there exists a constant β∈(0,1] such that
(2.22)∑n=1∞∑i=1nE(|Xni|β|an|β+|Xni|β)<∞,
then (2.4) holds true.Proof.
In Theorem2.4, we take
(2.23)gn(t)=|t|β1+|t|β,0<β≤1,n≥1.
It is easy to check that {gn(t),n≥1} is a sequence of nonnegative, even functions such that gn(|t|) is an increasing function of |t| for every n≥1. And
(2.24)gn(t)≥12tβ≥12t,0<t≤1,0<β≤1.
Therefore, by Theorem 2.4, we can easily get (2.4).



---

*Source: 430180-2011-07-21.xml*