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On some results of M. I. Gordin: A clarification of a misunderstanding. (English) Zbl 0645.60024
Let \(\{S_ n\}\) be the partial sums of a strictly stationary sequence \(\{X_ n\}\), where E \(X_ 0=0\) and \(E| X_ 0|^ p<\infty\) for some \(p>1\). For \(n\geq 1\), let \(\alpha (n)=\sup | P(AB)- P(A)P(B)|\), over \(A\in \sigma (...,X_{-1},X_ 0)\) and \(B\in \sigma (X_ n\), \(X_{n+1}\),...). If \[ \sum^{\infty}_{n=1}\alpha (n)^{1- 1/p}<\infty \quad and\quad \limsup \quad n^{-1/2}E| S_ n| <\infty, \] then M. I. Gordin [Abstracts of Communications, Int. Conf. Probability Theory and Mathematical Statistics, Vilnius, June 25- 30, p. 173-174 (1973)] proved that \(\lambda =\lim\) \(n^{-1/2}E| S_ n|\) exists and that \(n^{-1/2}S_ n\) converges in distribution to the normal distribution with mean zero and variance \(\pi \lambda^ 2/2.\)
The author points out that this result has been inadvertently misstated in a number of articles. He describes the chronology of events leading to this situation and discusses the subtle differences between the correct and incorrect formulations of the theorem.
Reviewer: R.J.Tomkins

MSC:
60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes
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