Limit theorems for stochastic exponentials of matrix-valued L\'evy processes

TL;DR

This paper investigates the asymptotic behavior of matrix-valued stochastic exponentials driven by Lévy processes, establishing laws of large numbers, central limit theorems, and Berry-Esseen bounds for key matrix characteristics.

Contribution

It provides new limit theorems and bounds for the long-term behavior of multiplicative Lévy processes in the general linear group.

Findings

Laws of large numbers for matrix logarithms

Central limit theorems for matrix entries and determinants

Berry-Esseen bounds where applicable

Abstract

We study the long-time behaviour of matrix-valued stochastic exponentials of L\'evy processes, i.e. of multiplicative L\'evy processes in the general linear group. In particular, we prove laws of large numbers as well as central limit theorems for the logarithmised norm, logarithmised entries and the logarithmised determinant of the stochastic exponential. Where possible, also Berry-Esseen bounds are stated.