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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
##### Keywords:
strong mixing; central limit theorem; strictly stationary
Full Text:
##### References:
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