# The n-term Approximation of Periodic Generalized L\'evy Processes

## TL;DR

This paper investigates the compressibility of generalized Lévy processes, driven by periodic Lévy white noises, revealing that non-Gaussian processes are more efficiently approximated in wavelet bases than Gaussian ones, with decay rates linked to the Lévy noise's Blumenthal-Getoor index.

## Contribution

It provides a novel analysis of the wavelet approximation efficiency of generalized Lévy processes, connecting their compressibility to the Lévy noise's Blumenthal-Getoor index.

## Key findings

- Non-Gaussian Lévy processes are more compressible than Gaussian ones.
- The decay rate of approximation error is quantified by the Blumenthal-Getoor index.
- Wavelet basis provides efficient n-term approximations for these processes.

## Abstract

In this paper, we study the compressibility of random processes and fields, called generalized L\'evy processes, that are solutions of stochastic differential equations driven by $d$-dimensional periodic L\'evy white noises. Our results are based on the estimation of the Besov regularity of L\'evy white noises and generalized L\'evy processes. We show in particular that non-Gaussian generalized L\'evy processes are more compressible in a wavelet basis than the corresponding Gaussian processes, in the sense that their $n$-term approximation error decays faster. We quantify this compressibility in terms of the Blumenthal-Getoor index of the underlying L\'evy white noise.