Hermitian Functional Representation of Free L\'evy Processes
Jos\'e-Luis P\'erez G., V\'ictor P\'erez-Abreu, Alfonso Rocha-Arteaga
TL;DR
This paper develops a functional representation for free Lévy processes using Hermitian matrix-valued Lévy processes, establishing asymptotic spectral behavior and connecting classical random matrix results to free probability theory.
Contribution
It introduces a novel functional framework for free Lévy processes and extends classical theorems to the free probability setting.
Findings
Functional asymptotics of spectral processes towards free Lévy laws
Recovery of a functional Wigner's theorem
Functional Marchenko-Pastur theorem with free Poisson limit
Abstract
A functional representation of free L\'evy processes is established via an ensemble of unitarily invariant Hermitian matrix-valued L\'evy processes. This is accomplished by proving functional asymptotics of their empirical spectral processes towards the law of a free L\'evy processes. This result recovers a functional version of Wigner's theorem and introduces a functional version of Marchenko-Pastur's theorem providing the free Poisson process as the noncommutative limit process.
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Taxonomy
TopicsRandom Matrices and Applications · Stochastic processes and statistical mechanics · Probability and Risk Models