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Abstract  

Sufficient conditions of covariance type are presented for weighted averages of random variables with arbitrary dependence structure to converge to 0, both for logarithmic and general weighting. As an application, an a.s. local limit theorem of Csáki, Földes and Révész is revisited and slightly improved.

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Abstract

Let {Xn: n ≧ 1} be a sequence of dependent random variables and let {wnk: 1 ≦ kn, n ≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.
limn1lognk=1n1kI(i=1kwkiXix)=12πxe12t2dta.s..
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sums, Acta Math. Hungar. 99 (4) (2003), 285-303. MR 2004a :60071 Almost sure convergence of weighted partial sums Acta Math. Hungar. 99

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