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On the Strong Forms of the Borel–Cantelli Lemma and Dynamical Systems with Polynomial Decay of Correlations

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Abstract

The strong forms of the Borel–Cantelli lemma are variants of the strong law of large numbers for sums of event indicators while the series of event probabilities diverges. These sums are centered at means and normalized by a certain function of means. In this paper, we derive new strong forms of the Borel–Cantelli lemma under wider restrictions on variations in the sum increments than has been done before. The strong forms are commonly used in analyzing the properties of dynamical systems. We apply our results to describe the properties of several measure-preserving expanding maps of [0, 1] with a fixed point at zero. Such results can also be proved in the case of analogous multidimensional maps.

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ACKNOWLEDGEMENTS

I cordially thank the anonymous reviewers for a number of valuable remarks that contributed to improving the quality of the text.

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Authors and Affiliations

  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    A. N. Frolov

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  1. A. N. Frolov
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Correspondence to A. N. Frolov.

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Translated by A. Shishulin

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Frolov, A.N. On the Strong Forms of the Borel–Cantelli Lemma and Dynamical Systems with Polynomial Decay of Correlations. Vestnik St.Petersb. Univ.Math. 55, 419–425 (2022). https://doi.org/10.1134/S1063454122040057

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  • DOI: https://doi.org/10.1134/S1063454122040057

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