  Let (X k) be a strictly stationary sequence of random variables with values
in some Polish space E and common marginal $\mu$, and (A k) k>0 be a sequence
of Borel sets in E. In this paper, we give some conditions on (X k) and (A k)
under which the events {X k $\in$ A k } satisfy the Borel-Cantelli (or strong
Borel-Cantelli) property. In particular we prove that, if $\mu$(lim sup n A n)
> 0, the Borel-Cantelli property holds for any absolutely regular sequence. In
case where the A k 's are nested, we show, on some examples, that a rate of
convergence of the mixing coefficients is needed. Finally we give extensions of
these results to weaker notions of dependence, yielding applications to
non-irreducible Markov chains and dynamical systems.
