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Abstract <p>New versions of the dynamic Borel<i>–</i>Cantelli lemma have been obtained for a sequence of nonnegative random variables without the assumption that they are uniformly bounded. Some alternating measure-preserving maps of the interval [0, 1] have been considered. In this case, the decay rates of respective random variables can be power-law distributed, and the correlations can be nonsummable. Estimates of these rates are known for random variables with supports of the interval (1/2, 1]. Therefore, the results do not follow from the known forms of the strong Borel<i>–</i>Cantelli lemma. The use of structures of &#xa0;intermittent maps makes it possible to obtain results for random variables with supports of (<i>d</i>, 1] for any <i>d</i> &gt; 0. These results are optimal, since there are examples showing that it is impossible to set <i>d</i> = 0.</p>

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On the Borel–Cantelli Lemma for Intermittent Interval Maps

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • A. N. Frolov

摘要

Abstract

New versions of the dynamic Borel–Cantelli lemma have been obtained for a sequence of nonnegative random variables without the assumption that they are uniformly bounded. Some alternating measure-preserving maps of the interval [0, 1] have been considered. In this case, the decay rates of respective random variables can be power-law distributed, and the correlations can be nonsummable. Estimates of these rates are known for random variables with supports of the interval (1/2, 1]. Therefore, the results do not follow from the known forms of the strong Borel–Cantelli lemma. The use of structures of  intermittent maps makes it possible to obtain results for random variables with supports of (d, 1] for any d > 0. These results are optimal, since there are examples showing that it is impossible to set d = 0.