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

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...

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Bibliographic Details
Published in:Vestnik, St. Petersburg University. Mathematics St. Petersburg University. Mathematics, 2022-12, Vol.55 (4), p.419-425
Main Author: Frolov, A. N.
Format: Article
Language:English
Subjects:
Analysis
Dynamical systems
Mathematics
Mathematics and Statistics
Polynomials
Probability theory
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Summary: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.
ISSN:1063-4541
1934-7855
DOI:10.1134/S1063454122040057