## On the central limit theorem for $$\rho$$-mixing sequences of random variables.(English)Zbl 0638.60032

Let $$...X_{-1},X_ 0,X_ 1,..$$. denote a strictly stationary sequence of random variables with E $$X_ 0=0$$, E X$${}$$ $$2_ 0<\infty$$ and set $$S_ n=X_ 1+...+X_ n$$, $$\sigma$$ $$2_ n=Var S_ n$$. Assume that $$\sigma_ n\to \infty$$ as $$n\to \infty$$ and further that $$\rho$$ (n)$$\to 0$$ where $$\rho$$ is the dependence coefficient defined by $$\rho (n)=\sup | corr(f,g)|$$, with the supremum being taken over all square- integrable f, g with f measurable with respect to $$\sigma \{X_ k$$, $$k\leq 0\}$$ and g with respect to $$\sigma \{X_ k$$, $$k\geq n\}.$$
I. A. Ibragimov [Theor. Veroyatn. Primen. 20, 134-140 (1975; Zbl 0335.60023); English translation in Theor. Probab. Appl. 20, 135-141 (1975)] showed that if either (i) E $$| X_ 0|^{2+\delta}<\infty$$, for some $$\delta >0$$, or (ii) $$\sum^{\infty}_{1}\rho (2$$ $$i)<\infty$$ then $$S_ n/\sigma_ n\Rightarrow N(0,1)$$ as $$n\to \infty$$, where $$\Rightarrow$$ denotes convergence in distribution.
The principal result of this paper is that if there exists a function g with E X$${}$$ $$2_ 0 g(| X_ 0|)<\infty$$ and $g(n^{1/2})=O(\exp (2\sum^{[\log n]}_{1}\rho (2\quad i)/(1- \mu)),\text{ for some } 0<\mu <1,$ then $$S_ n/\sigma_ n\Rightarrow N(0,1)$$. These conditions reduce to (i) if $$g(x)=x^{\delta}$$, and to (ii) if g is constant.
Reviewer: D.P.Kennedy

### MSC:

 60F05 Central limit and other weak theorems 60B10 Convergence of probability measures 60G10 Stationary stochastic processes

Zbl 0335.60023
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