Repetition and recurrence times: Dual statements and short memory conditions

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 condition for the upper bound resembles short memory in the sense developed in time series analysis. Our reasonings make 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.