The Shortest Possible Return Time of β-Mixing Processes

Miguel Abadi; Sandro Gallo and Erika Alejandra Rada-Mora

We consider a stochastic process and a givenn-string. We study the shortestpossiblereturn time (or shortest return path) of the string over all the realizations of process starting from this string. For aβ-mixing process having complete grammar, and for each sizenof the strings, we approximate the distribution of this short return (properly re-scaled) by a non-degenerated distribution. Under mild conditions on theβcoefficients, 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.

The whole paper is available here.

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