Quantitative and stable limits of high-frequency statistics of L\'evy processes: a Stein's method approach

TL;DR

This paper develops a Stein's method approach to quantify the distributional distance and establish stable convergence of high-frequency statistics of multidimensional Lévy processes to mixed Gaussian distributions.

Contribution

It introduces a novel Stein's method framework tailored for high-frequency Lévy process statistics, providing new inequalities and convergence results.

Findings

Derived inequalities for distributional distance between errors and mixed Gaussian variables

Established stable functional convergence for high-frequency statistics

Adapted Stein's method to the context of Lévy processes and high-frequency data

Abstract

We establish inequalities for assessing the distance between the distribution of errors of partially observed high-frequency statistics of multidimensional L\'evy processes and that of a mixed Gaussian random variable. Furthermore, we provide a general result guaranteeing stable functional convergence. Our arguments rely on a suitable adaptation of the Stein's method perspective to the context of mixed Gaussian distributions, specifically tailored to the framework of high-frequency statistics.