On the Marcinkiewicz–Zygmund strong laws for arbitrary dependent sequences

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Abstract

Strong laws of large numbers are obtained for non-homogeneous arbitrary dependent sequences.

Section snippets

Result

Let Yk be a sequence of non-negative random variables defined on a probability space (Ω,F,P) and set Rn=max1knYk. The main result is as follows.

Theorem 1

Suppose bn is non-negative and non-decreasing sequence such that bn1Rn0 in Lq,q>0. If k=11bkq1bk+1qE(Rkq)<then bn1Rn converges almost surely. Moreover, if lim supnnbn+1qbnq1<then, (1.1) can be replaced by k=1E(Rkq)kbkq<.

Let {Xk}, k=1,2, be a random sequence defined on (Ω,F,P) and set Sn=k=1nXk, Zn=max1kn|Sk|. Let h be a slowly varying

Proof

We have the following Hájek–Rényi type inequality (cf. Hájek and Rényi, 1955).

Theorem 2

Let c1c2cn0, n1. Then, for any m, nm1 and x>0, >q>0 xqP(maxmknckYk>x)k=mn1(ckqck+1q)E(Rkq)+cnqE(Rnq).

Proof of Theorem 2

Assume m=1. For ν=1,2,,n set Cν={c1Y1x,,cν1Yν1x,cνYν>x}.Clearly C=1nCν={max1knckYk>x}. Suppose first that q1, then xqP(C)=ν=1nk=νn1(ckqck+1q)xqcνqP(Cν)+ν=1ncnqxqcνqP(Cν)ν=1nk=νn1(ckqck+1q)E(YνqICν)+ν=1ncnqE(YnqICν)ν=1nk=νn1(ckqck+1q)E(RkqICν)+ν=1ncnqE(RnqICν)ν=1nk=1n1(ckqc

Application

Suppose f(x)=x1ph(x), p(0,1) and h is slowly varying. Without loss of generality we can assume that f is increasing and differentiable (cf. Proposition B1.9(3), p.366 in de Haan and Fereira, 2006). Define the generalized inverse of f g(y)=f(y)inf{xR+|f(x)y}, inf. Further, g(y)=xph(y), where h is a slowly varying function, with f(g(x))=g(f(x))=x, x[A,), for some A0 (cf. Proposition B1.9(9), p.367 in de Haan and Fereira, 2006). Assume that random variables Xk, k=1,2 are

Acknowledgments

I would like to thank the editor and referee for comments and suggestions.

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