Entropic Compressibility of L\'evy Processes

TL;DR

This paper investigates the compressibility of L\'evy processes by analyzing the entropy of their discretized versions, extending known results to non-stable cases and introducing a hierarchy based on their Blumenthal-Getoor index.

Contribution

It introduces a new framework for understanding the entropy and compressibility of L\'evy processes, including a local central limit theorem and a hierarchy based on the Blumenthal-Getoor index.

Findings

Derived asymptotics of differential entropy for small-time marginals.

Extended results from stable to non-stable L\'evy processes.

Established a hierarchy of processes based on compressibility.

Abstract

In contrast to their seemingly simple and shared structure of independence and stationarity, L\'evy 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 (2018), 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 L\'evy 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 L\'evy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of L\'evy processes, captured by their Blumenthal-Getoor index.