On strong forms of the Borel-Cantelli lemma and dynamical systems with polynomial decays of correlations

Strong forms of the Borel - Cantelli lemma are variants strong law large numbers for sums indicators events such that series from probabilities these diverges. These centered at means and normalized by some function means. In this paper, we derive new under wider restrictions on variations increments than it was done earlier. commonly used investigations properties dynamical systems. We apply our results to describe measure preserving expanding maps [0, 1] with a fixed point zero. Such can be proved similar multidimensional as well. Keywords: lemma, quantitative events, num bers, almost surely convergence, systems, polynomial decay correlations.