  We derive new variants of the quantitative Borel--Cantelli lemma and apply
them to analysis of statistical properties for some dynamical systems. We
consider intermittent maps of $(0,1]$ which have absolutely continuous
invariant probability measures. In particular, we prove that every sequence of
intervals with left endpoints uniformly separated from zero is the strong
Borel--Cantelli sequence with respect to such map and invariant measure.
