On the Borel-Cantelli lemma and its dynamical forms for transformations of intervals

The paper is devoted to estimating of the growth rate for sums of non-negative measurable functions. Such estimates are of significant interest in probability theory and the theory of dynamical systems. In this paper, new versions of the strong Borel-Cantelli lemma for non-negative random variables are obtained. The random variables are not assumed to be uniformly bounded. The new versions of the strong Borel-Cantelli lemma are stronger than earlier results for indicators of events as well. The obtained results are used to describe the statistical properties of dynamical systems. Some measure-preserving transformations of the interval $[0, 1]$ are considered. The rate of decay of correlations of random variables in the dynamical systems studied in the paper is exponential. New versions of the dynamical Borel-Cantelli lemma are proved. In the paper, there are two variants of conditions on covariances which lead to results with different normalizing sequences. It is shown that if the number of large random variables in the sequence is small enough, then it is possible to choose smaller normalizing constants and, therefore, the result will be better. Relevant examples are given.