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