Abstract <p>This work is devoted to estimating the growth rate for sums of nonnegative measurable functions. Such estimates are of significant interest in probability theory and the theory of dynamical systems. In this work, we derive new versions of the strong Borel–Cantelli lemma for nonnegative random variables. 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 maps of the interval [0, 1] are considered. The rate of decay of correlations of random variables in the dynamical systems studied in the work is exponential. The new versions of the dynamical Borel–Cantelli lemma are proved. The paper presents two variants of conditions on the 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 we can choose smaller normalizing constants and, therefore, make the result better. Relevant examples are given.</p>