Repetition and recurrence times: Dual statements and summable mixing rates

Abstract

By an analogy to the duality between the recurrence time and the longest match length, we introduce a quantity dual to the maximal repetition length, which we call the repetition time. Extending prior results, we sandwich the repetition time in terms of unconditional and conditional min-entropies. The upper bound holds if the mixing rate $ϕ(n)$ is summable, whereas the lower bound only assumes stationarity. Our reasoning makes a repeated use of dualities between so-called times and so-called counts that generalize the duality of the recurrence time and the longest match length. We also discuss the analogy of these results with the Wyner-Ziv/Ornstein-Weiss theorem, which sandwiches the recurrence time in terms of Shannon entropies.