Weak convergence of the empirical truncated distribution function of the L\'evy measure of an It\=o semimartingale

TL;DR

This paper proves that a normalized empirical distribution function of the Lévy measure for high-frequency observed Itô semimartingales converges weakly to a Gaussian process, even with a non-zero diffusion component.

Contribution

It introduces a new estimator for the Lévy measure's distribution function that is applicable to processes with diffusion components and simple jump process assumptions.

Findings

Estimator converges weakly to a Gaussian process.

Works for processes with non-vanishing diffusion components.

Operates under simple jump process assumptions.

Abstract

Given an It\=o semimartingale with a time-homogeneous jump part observed at high frequency, we prove weak convergence of a normalized truncated empirical distribution function of the L\'evy measure to a Gaussian process. In contrast to competing procedures, our estimator works for processes with a non-vanishing diffusion component and under simple assumptions on the jump process.