Entropic Compressibility of Lévy Processes

Abstract

In contrast to their seemingly simple and shared structure of independence and stationarity, Lévy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes. Inspired by the recent paper of Ghourchian, Amini, and Gohari [32], we characterize their compressibility by studying the entropy of their double discretization (both in time and amplitude) in the regime of vanishing discretization steps. For a Lévy process with absolutely continuous marginals, this reduces to understanding the asymptotics of the differential entropy of its marginals at small times. We generalize known results for stable processes to the non-stable case, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal-Getoor index.