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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

Citations:

Zbl 0335.60023
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