Compound Poisson statistics for multiple returns in shrinking cylinders for mixing processes

Abstract

Given a periodic point ${\it\omega}$ in a ${\it\psi}$-mixing shift with countable alphabet, the sequence $\{S_{n}\}$ of random variables counting the number of multiple returns in shrinking cylindrical neighborhoods of ${\it\omega}$ 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 Pólya–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 also exhibit an example for which the limit distribution is pure Poissonian.