The Shortest Possible Return Time of $\beta$ -Mixing Processes

We consider a stochastic process and a given n-string. We study the shortest possible return time (or shortest return path) of the string over all the realizations of process starting from this string. For a β-mixing process having complete grammar, and for each size n of the strings, we approximate the distribution of this short return (properly re-scaled) by a non-degenerated distribution. Under mild conditions on the β coefficients, we prove the existence of the limit of this distribution to a non-degenerated distribution. We also prove that ergodicity is not enough to guaranty this convergence. Finally, we present a connection between the shortest return and the Shannon entropy, showing that maximum of the re-scaled variables grow as the matching function of Wyner and Ziv.