On complete convergence for arrays of rowwise dependent random variables

https://doi.org/10.1016/j.spl.2006.12.007Get rights and content

Abstract

This paper establishes two results for complete convergence in the law of large numbers for arrays under ϱ-mixing and ϱ˜-mixing association in rows. They extend several known results.

Introduction

Let {Xn,n1} be a sequence of random variables defined on probability space (Ω,F,P). A sequence {Xn,n1} is said to converge completely to a constant C if n=1P[|Xn-C|>ɛ]<,ɛ>0.Hsu and Robbins (1947), who introduce this concept, proved that the sequence of arithmetic means of independent identically distributed random variables converges completely to the expected value of the summands, provided the variance is finite. The converse theorem was proved by Erdös (1949).

The extensions of Hsu–Robbins–Erdös's result, due to Katz (1963), Baum and Katz (1965), Chow (1973), form a complete convergence theorem with a Marcinkiewicz–Zygmunt type normalization (see Gut, 1983).

Theorem 1.1

Let {Xn,n1} be a sequence of independent identically distributed random variables and let αr1, α>12. The following statements are equivalent: (i)E|X1|r<and ifr1,EX1=0,(ii)n=1nαr-2Pi=1nXi>nαɛ<,ɛ>0,(iii)n=1nαr-2Pmaxkni=1kXi>nαɛ<,ɛ>0.If αr>1, α>12 the above are also equivalent to (iv)n=1nαr-2Psupknk-αi=1kXi|>ɛ<,ɛ>0.

Many authors generalized and extended this result without assumption of identical distribution in several directions. They studied the cases of independent, stochastically dominated random variables, triangular arrays of rowwise independent, stochastically dominated in the Cesaro sense random variables and sequences of independent random variables taking value in a Banach space (Pruitt, 1966, Rohatgi, 1971, Hu et al., 1989, Gut, 1992, Wang et al., 1993; Kuczmaszewska and Szynal, 1988, Kuczmaszewska and Szynal, 1991, Kuczmaszewska and Szynal, 1994; Sung, 1997, Bozorgnia et al., 1993, Hu et al., 1999; Hu et al., 1998, Hu and Volodin, 2000, Ahmed et al., 2002, Kuczmaszewska, 2004, Sung et al., 2005; Tómács, 2005).

In this paper, we consider a complete convergence in the strong law of large numbers for arrays of dependent random variables. We study the complete convergence for ϱ-mixing and ϱ˜-mixing sequences of random variables. The obtained results extend some previous known. Some results for complete convergence for ϱ-mixing and ϱ˜-mixing random variables one can find Zhengyan and Chuanrong (1996) (Shao, Kong and Zang, Section 8.4), Shao (1995), Shixin (2004).

Definition 1.1

A sequence of random variables {Xn,n1} is said to be a ϱ-mixing sequence if the maximal correlation coefficientϱ(n)=supk1,XL2(F1k),YL2(Fk+n)|cov(X,Y)|VarX·VarY0,as n, where Fnm is the σ-field generated by random variables Xn,Xn+1,, Xm.

Definition 1.2

A sequence of random variable {Xn,n1} is said to be a ϱ˜-mixing sequence if there exists kN such thatϱ˜(k)=supS,TsupXL2(FS),YL2(FT)cov(X,Y)VarX·VarY<1,where S, T are the finite subsets of positive integers such that dist (S,T)k and FW is the σ-field generated by the random variable {Xi,iWN}.

The ϱ˜-mixing conception is similar to the ϱ-mixing, but they are quite different from each other.

In our further consideration we need the following definition and lemmas.

Definition 1.3

A real valued function l(x), positive and measurable on [A,) for some A>0, is said to be slowly varying if limxl(x·λ)l(x)=1foreachλ>0.

Lemma 1.1 Shao, 1995

Let {Xn,n1} be a ϱ-mixing sequence of random variables such that EXn=0, n1. Then for any q2, there exists a constant K=K(q,ϱ(·)) depending only on q and ϱ(·) such that:Emaxkn|Sk|qKnq/2expKi=0[lnn]ϱ(2i)maxkn(EXk2)q/2+nexpKi=0[lnn]ϱ2/q(2i)maxknE|Xk|q.

Lemma 1.2 Shanchao, 1998; Peligrad and Gut, 1999

Let {ξn,n1} be a ϱ˜-mixing sequence with ϱ˜(1)<1. Let Xnσ(ξi,i1), EXn=0, E|Xn|p<, n1, p>1. Then there exists a positive constant C such that:Ei=1nXiqCi=1nE|Xi|q,n1,1<q2,Ei=1nXiqCi=1nE|Xi|q+i=1nEXi2q2,n1,q>2.

We also assume that in our consideration constant C is not the same constant in each case.

Section snippets

The main results

In this paper we consider arrays of random variables.

Let ϱn(i) denotes the maximal correlation coefficient defined in (1.1) for the nth row of an array {Xni,i1,n1} i.e. for the sequence Xn1,Xn2,Xn3,, n1.

Similarly, we will use the notation ϱ˜n(k) for denoting the coefficient defined in (1.2) for the sequence Xn1,Xn2,Xn3,..., n1.

Moreover, let {ϕn,n1} be a sequence of nonnegative, even, continuous and nondecreasing on (0,) functions ϕn:RR+ with limxϕn(x)= and such that:ϕn(x)/xandϕn(x)/x

Acknowledgment

The author is very grateful to the referee for the suggestions which allowed her to improve this paper.

References (27)

  • A. Gut

    Complete convergence and convergence rates for randomly indexed partial sums with an application to some first passage times

    Acta Math. Hungar.

    (1983)
  • A. Gut

    Complete convergence for arrays

    Periodica Math. Hungar.

    (1992)
  • P.L. Hsu et al.

    Complete convergence and the law of large numbers

    Proc. Nat. Acad. Sci. USA

    (1947)
  • Cited by (19)

    View all citing articles on Scopus
    View full text