Limit Theorems for Stochastic Exponentials of Matrix-Valued Lévy Processes

Anita Behme () and Sebastian Mentemeier ()
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Anita Behme: Technische Universität Dresden
Sebastian Mentemeier: Universität Hildesheim

Journal of Theoretical Probability, 2025, vol. 38, issue 3, 1-40

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

Keywords: Central limit theorem; Law of large numbers; Multivariate Lévy process; Multiplicative Lévy process; Products of random matrices; Stochastic exponential; Lévy processes on groups; Primary 60G51; 60J57; Secondary 60H10; 60B15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01430-z

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