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On moment conditions for normed sums of independent variables and martingale differences. (English) Zbl 0554.60050

This article presents several criteria so that \(\{n^{- 1/q}\sum^{n}_{k=1}X_ k\}\) will have uniformly bounded p-th moments, for \(0<p<q\leq 2\), where \(\{X_ n\}\) is a sequence of i.i.d. r.v’s. These results are extended to an ergodic sequence of martingale differences; more precisely, it is shown that \(E| n^{-1/2}\sum^{n}_{k=1}X_ k|\) converges to a finite limit iff \(EX^ 2_ k<\infty\) for all k. The authors’ proof for this martingale difference sequence corrects an inaccuracy in the proof presented by P. Hall and C. C. Heyde in their book ”Martingale limit theory and its application” (1980; Zbl 0462.60045).
Reviewer: D.Kannan

MSC:

60G50 Sums of independent random variables; random walks
60G42 Martingales with discrete parameter
60F25 \(L^p\)-limit theorems

Citations:

Zbl 0462.60045
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References:

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