🤖 AI Summary
This paper addresses the quantification of repetitiveness in time series by introducing the novel concept of “repetition time”—a dual quantity to “maximum repetition length”—and establishing a “time–count” duality framework. Methodologically, it leverages information theory and ergodic theory to derive tight upper and lower bounds on repetition time via unconditional and conditional minimum entropy. The analysis reveals an intrinsic connection between repetition time and short-memory properties. The work extends the Ornstein–Weiss-type entropy bound theory by rigorously identifying the decisive role of the short-memory condition in repetitiveness measurement. Furthermore, it unifies the symmetric structural characterization of three fundamental quantities—recurrence time, longest match length, and maximum repetition length—thereby providing a new theoretical foundation and analytical toolkit for studying repetitiveness in short-memory stochastic processes.
📝 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.
