The Annals of Probability

Classical and free infinitely divisible distributions and random matrices

Florent Benaych-Georges

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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).

Article information

Ann. Probab., Volume 33, Number 3 (2005), 1134-1170.

First available in Project Euclid: 6 May 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52 46L54: Free probability and free operator algebras
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems

Random matrices free probability asymptotic freeness free convolution Marchenko–Pastur distribution infinitely divisible distributions


Benaych-Georges, Florent. Classical and free infinitely divisible distributions and random matrices. Ann. Probab. 33 (2005), no. 3, 1134--1170. doi:10.1214/009117904000000982.

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