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On the central limit question under absolute regularity. (English) Zbl 0582.60047
Let $$X=(X_ k,k\in {\mathbb{Z}})$$ be a strictly stationary sequence of random variables on a probability space ($$\Omega$$,$${\mathfrak F},P)$$. Denote $$S_ n=X_ 1+...+X_ n$$, $$n=1,2,..$$. $$\beta ({\mathfrak A},{\mathfrak B})=\sup \quad 2^{-1}\sum^{I}_{i=1}\sum^{J}_{j=1}| P(A_ i\cap B_ j)- P(A_ i)P(B_ j)|,$$ where sup is taken over all pairs of partitions $$\{A_ 1,...,A_ I\}$$ and $$\{B_ 1,...,B_ J\}$$ of $$\Omega$$ such that $$A_ i\in {\mathfrak A}$$ $$\forall i$$ and $$B_ j\in {\mathfrak B}$$ $$\forall j$$; $${\mathfrak A}$$, $${\mathfrak B}$$ being $$\sigma$$-fields. $$\beta (n)=\beta ({\mathfrak F}^ 0_{-\infty},{\mathfrak F}_ n^{\infty})$$, $$n=1,2,..$$. Let (1) $$0<Var X_ 0<\infty$$, $$corr(X_ 0,X_ n)=0$$ $$\forall n\geq 1$$; (2) $$\inf_{n\geq 1}P(S_ n=0)>0$$; (3) $$\lim_{c\to \infty}[\sup_{n\geq 1}P(| S_ n| >c)]=0.$$
Theorem 3. Suppose $$\delta >0$$. Then there exists a strictly stationary sequence $$X=(X_ k)$$ such that $$EX_ 0=0$$, $$E| X_ 0|^{2+\delta}<\infty$$, $$\sum^{\infty}_{m=1}\beta (m)^{\delta /2+\delta}<\infty$$, Var $$S_ m\sim (\log m)^{-4}\cdot m$$ as $$m\to \infty$$ and (2) and (3) both hold.
Theorem 4. There exists a strictly stationary sequence $$X=(X_ k)$$ such that $$EX_ 0=0$$, $$| X_ 0| <C$$ a.s. for some $$C<\infty$$, $$\sum^{\infty}_{m=1}\beta (m)<\infty$$, Var $$S_ m\sim (\log m)^{- 4}\cdot m$$ as $$m\to \infty$$, and (2) and (3) both hold.
Reviewer: M.Mirzahmedov

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