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Given a periodic point $\omega$ in a $\psi$-mixing shift with countablealphabet, the sequence $\{S_{n}\}$ of random variables counting the number ofmultiple returns to shrinking cylindrical neighborhoods of $\omega$ isconsidered. Necessary and sufficient conditions for the convergence indistribution of $\{S_{n}\}$ are obtained, and it is shown that the limit is aPolya-Aeppli distribution. A global condition on the shift system, whichguarantees the convergence in distribution of $\{S_{n}\}$ for every periodicpoint, is introduced. This condition is used to derive results for$f$-expansions and Gibbs measures. Results are also obtained concerning thepossible limit distribution of sub-sequences $\{S_{n_{k}}\}$. A family ofexamples in which there is no convergence is presented. We exhibit also anexample for which the limit distribution is pure Poissonian.

Compound Poisson statistics in conventional and nonconventional setups