Total variation distance for discretely observed Lévy processes: A Gaussian approximation of the small jumps

Abstract

It is common practice to treat small jumps of Levy 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 Levy processes with infinite Levy measure in total variation distance. Non asymptotic bounds for the total variation distance between n discrete observations of small jumps of a Levy 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 Levy processes are provided. The theory is illustrated by concrete examples.