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.

We define the first match function Tn : C n → {1, ... , n} where C is a finite alphabet. For two copies of x 1 n ∈ C n , this function gives the minimum number of steps one has to slide one copy of x 1 n to get a match with the other one. For ergodic positive entropy processes, Saussol and coauthors proved the almost sure convergence of T n /n. We compute the large deviation properties of this function. We prove that this limit is related to the Rényi entropy function, which is also proved to exist. Our results hold under a condition easy to check which defines a large class of processes. We provide some examples.