A Gaussian approximation theorem for L\'evy processes

TL;DR

This paper proves a Gaussian approximation theorem for Lévy processes, showing how their distributions converge to a normal distribution without requiring higher moments, extending classical results to continuous time.

Contribution

It introduces a new Gaussian approximation theorem for Lévy processes that does not rely on higher moment assumptions, extending classical CLT results to continuous-time processes.

Findings

Decay of Kolmogorov distance established

Extension of classical CLT to Lévy processes

No higher moment assumptions needed

Abstract

Without higher moment assumptions, this note establishes the decay of the Kolmogorov distance in a central limit theorem for L\'evy processes. This theorem can be viewed as a continuous-time extension of the classical random walk result by Friedman, Katz and Koopmans.