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Entropic Compressibility of Lévy Processes

Julien Fageot, Alireza Fallah, Thibaut Horel

TL;DR

This work characterize their compressibility by studying the entropy of their double discretization (both in time and amplitude) in the regime of vanishing discretized steps, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal–Getoor index.

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 (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 Lévy 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 Lévy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal-Getoor index.

Entropic Compressibility of Lévy Processes

TL;DR

This work characterize their compressibility by studying the entropy of their double discretization (both in time and amplitude) in the regime of vanishing discretized steps, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal–Getoor index.

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 (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 Lévy 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 Lévy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal-Getoor index.

Paper Structure

This paper contains 24 sections, 14 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions: the Asymptotic Entropy of Lévy Processes
  3. Related Work
  4. Outline
  5. Lévy Processes and their Blumenthal--Getoor Index
  6. Lévy Processes
  7. Symmetric-alpha-Stable Lévy Processes
  8. The Blumenthal--Getoor Index of Lévy Processes
  9. Entropy of Lévy Processes
  10. Quantization and Entropy of Random Variables
  11. Definition and Existence of the Entropy of Lévy Processes
  12. Entropy of Lévy Processes at Small Times
  13. Locally Self-Similar Lévy Processes
  14. A Small-time Limit Theorem for the Entropy of LSS Lévy Processes
  15. An Upper Bound on the Small-Time Entropy of Lévy Processes
  16. ...and 9 more sections

Key Result

Proposition 3.2

Let $X$ be a real random variable. If $\mathop{\mathrm{H}}\nolimits(\lfloor X\rfloor)<\infty$, then $\mathop{\mathrm{H}}\nolimits([X]_m)<\infty$ for all $m\geq 1$. If furthermore $X$ has finite differential entropy, then

Figures (1)

  • Figure 1: Entropic compressibility of Lévy processes with absolutely continuous marginals. The expressions shown are upper-bounds for general Lévy processes and equal to the dominant term of the asymptotics of $h(L_{t})$ for locally symmetric and self-similar processes. Wiener processes and S$\alpha$S stable processes were already covered in Ghourchian2017compressible.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Remark
  • Definition 3.1
  • Proposition 3.2: renyi1959dimension
  • Remark
  • Proposition 3.3
  • Definition 3.4
  • Remark
  • Proposition 3.5
  • ...and 35 more