Entropic Compressibility of Levy Processes

In contrast to their seemingly simple and shared structure of independence and stationarity, Levy 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, 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 Levy process with absolutely continuous marginals, this reduces to understanding the asymptotics of the differential entropy of its marginals at small times, for which we obtain a new local central limit theorem. We generalize known results for stable processes to the non-stable case, with a special focus on Levy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of Levy processes, captured by their Blumenthal-Getoor index.