Total variation distance for discretely observed L\'evy processes: a Gaussian approximation of the small jumps

TL;DR

This paper investigates the accuracy of approximating small jumps of Le9vy processes by Gaussian distributions using total variation distance, providing non-asymptotic bounds and practical examples.

Contribution

It offers the first detailed analysis of Gaussian approximation for small jumps of Le9vy processes in total variation distance with explicit bounds.

Findings

Non-asymptotic bounds for total variation distance between observations and Gaussian approximation

New upper bounds for total variation distance between Le9vy process observations

Illustrative examples demonstrating the theory

Abstract

It is common practice to treat small jumps of L\'evy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of L\'evy processes with infinite L\'evy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a L\'evy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of L\'evy processes are provided. The theory is illustrated by concrete examples.