  Given a periodic point $\omega$ in a $\psi$-mixing shift with countable
alphabet, the sequence $\{S_{n}\}$ of random variables counting the number of
multiple returns to shrinking cylindrical neighborhoods of $\omega$ is
considered. Necessary and sufficient conditions for the convergence in
distribution of $\{S_{n}\}$ are obtained, and it is shown that the limit is a
Polya-Aeppli distribution. A global condition on the shift system, which
guarantees the convergence in distribution of $\{S_{n}\}$ for every periodic
point, is introduced. This condition is used to derive results for
$f$-expansions and Gibbs measures. Results are also obtained concerning the
possible limit distribution of sub-sequences $\{S_{n_{k}}\}$. A family of
examples in which there is no convergence is presented. We exhibit also an
example for which the limit distribution is pure Poissonian.
