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May 2005 Classical and free infinitely divisible distributions and random matrices
Florent Benaych-Georges
Ann. Probab. 33(3): 1134-1170 (May 2005). DOI: 10.1214/009117904000000982

Abstract

We construct a random matrix model for the bijection Ψ between clas- sical and free infinitely divisible distributions: for every d≥1, we associate in a quite natural way to each *-infinitely divisible distribution μ a distribution ℙdμ on the space of d×d Hermitian matrices such that ℙdμ*ℙdν=ℙdμ*ν. The spectral distribution of a random matrix with distribution ℙdμ converges in probability to Ψ(μ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko–Pastur distribution. In an analogous way, for every d≥1, we associate to each *-infinitely divisible distribution μ, a distribution $\mathbb{L}_{d}^{\mu}$ on the space of complex (non-Hermitian) d×d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |Md|, when Md is $\mathbb{L}_{d}^{\mu}$-distributed, converges in probability to Ψ(μ).

Citation

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Florent Benaych-Georges. "Classical and free infinitely divisible distributions and random matrices." Ann. Probab. 33 (3) 1134 - 1170, May 2005. https://doi.org/10.1214/009117904000000982

Information

Published: May 2005
First available in Project Euclid: 6 May 2005

zbMATH: 1076.15022
MathSciNet: MR2135315
Digital Object Identifier: 10.1214/009117904000000982

Subjects:
Primary: 15A52 , 46L54
Secondary: 60E07 , 60F05

Keywords: asymptotic freeness , Free convolution , Free probability , Infinitely divisible distributions , Marchenko–Pastur distribution , random matrices

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 3 • May 2005
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