On moment conditions for normed sums of independent variables and martingale differences (Q760093)

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 \textit{P. Hall} and \textit{C. C. Heyde} in their book ''Martingale limit theory and its application'' (1980; Zbl 0462.60045).