Random matrix models of stochastic integral type for free infinitely divisible distributions
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RANDOM MATRIX MODELS OF STOCHASTIC INTEGRAL TYPE FOR FREE INFINITELY DIVISIBLE DISTRIBUTIONS J. Armando Dom´ınguez-Molina1 and Alfonso Rocha-Arteaga2 1
Escuela de Ciencias F´ısico-Matem´ aticas Universidad Aut´ onoma de Sinaloa, M´exico E-mail: [email protected] 2
Escuela de Ciencias F´ısico-Matem´ aticas Universidad Aut´ onoma de Sinaloa, M´exico E-mail: [email protected] (Received May 18, 2010; Accepted July 2, 2010) [Communicated by D´enes Petz]
Abstract The Bercovici–Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the L´evy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued L´evy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random R∞ matrix model of Ornstein–Uhlenbeck type 0 e−t dΨdt , d ≥ 1, where Ψdt is a d × d matrix-valued L´evy process satisfying an Ilog condition.
1. Introduction An ensemble of random matrices is a sequence (Md )d≥1 where Md is a d × d matrix whose entries are random variables. The spectral distribution of Md is the Mathematics subject classification numbers: 15B52, 46L54, 60E07, 60H05. Key words and phrases: infinitely divisible distribution, random matrices, stochastic integrals, polar decomposition, L´evy measures. 0031-5303/2012/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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J. A. DOM´INGUEZ-MOLINA and A. ROCHA-ARTEAGA
uniform distribution of its spectrum λ1 , λ2 , . . . , λd , that is the (random) probability Pd measure µMd defined as µMd = 1d i=1 δλi . A random matrix model for a probability measure µ is an ensemble (Md )d≥1 for which the spectral distribution µMd converges weakly to µ. Bercovici and Pata [7] introduced a bijection Λ from the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions to study relations between classical and free infinitely divisible aspects. Under this bijection Benaych-Georges [6] and Cavanal-Duvillard [8] construct for any classical one-dimensional infinitely divisible distribution µ a random matrix model for the corresponding free infinitely divisible distribution Λ (µ). This include the Wigner and Marchenko–Pastur results, which provide random matrix models of
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