Repetition and recurrence times: Dual statements and summable mixing rates

TL;DR

This paper introduces the concept of repetition time as a dual to maximal repetition length, providing bounds based on mixing rates and entropies, extending previous theoretical results in ergodic theory.

Contribution

It defines the repetition time and establishes bounds using mixing rates and entropies, generalizing duality principles in ergodic theory.

Findings

Repetition time is bounded by min-entropies under summable mixing rates.

Lower bounds hold for stationary processes without mixing assumptions.

Analogy with Wyner-Ziv/Ornstein-Weiss theorem is discussed.

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 $\phi(n)$ 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.