Refinement of the almost sure central limit theorem for associated processes (Q5942010)

The aim of this paper is to generalize the almost sure CLT and the limit theorem for logarithmic averages to the case of associated random sequences. More precisely, let \((X_n)\) be a zero-mean stationary associated random sequence, \(S_n=X_1+ \cdots+ X_n\) and \(Y_n=\sum^n_{i=1} {f(S_j/ \sqrt j) \over j}\), where \(f\) is a Lipschitz function. It is proved that \(Y_n/ \log_n\) converges almost surely to \(Ef(\xi)\) where \(\xi\) is a zero-mean normal random variable. Moreover, if additionally \(f\) is monotone and bounded, then \((Y_n-EY_n)/ \sqrt{\log n}\) converges weakly to the normal distribution. The second result generalizes and refines the the theorem of \textit{M. Csörgő} and \textit{L. Horváth} [Math. Proc. Camb. Philos. Soc. 112, No. 1, 195-205 (1992; Zbl 0766.60038)] which was obtained in the i.i.d. case.