A random matrix model for the Gaussian distribution
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A RANDOM MATRIX MODEL FOR THE GAUSSIAN DISTRIBUTION V´ıctor P´ erez-Abreu (Cuajimalpa) [Communicated by: D´enes Petz ] Departmento de Matem´ aticas Aplicadas y Sistemas Universidad Aut´ onoma Metropolitana-Cuajimalpa Prolongaci´ on Canal de Miramontes No. 3855 Col. Ex-Hacienda de San Juan de Dios Tlalpan 14387 D. F., M´exico E-mail: [email protected] (Received: April 27, 2005; Accepted: May 24, 2005)
Abstract The Gaussian unitary ensemble is a random matrix model (RMM) for the Wigner law. While random matrices in this model are infinitely divisible, the Wigner law is infinitely divisible not in the classical but in the free sense. We prove that any variance mixture of Gaussian distributions – whether infinitely divisible or not in the classical sense – admits a RMM of non Gaussian infinitely divisible random matrices. More generally, it is shown that any mixture of the Wigner law admits a RMM. A key role is played by the fact that the Gaussian distribution is the mixture of Wigner law with the 2-gamma distribution.
1. Introduction and notation An ensemble of random matrices is a sequence X d d≥1 where for each d ≥ 1, d
X d is a d × d matrix with random entries. The spectral measure µ X of X d is d d Xd Xd defined as the empirical distribution of its eigenvalues λ1 , . . . , λd , that is, µ X d = d d−1 i=1 δλX d . A probability measure µ is said to admit a random matrix model i
Mathematics subject classification number: Primary 60B15; Secondary 60F05. Key words and phrases: matrix valued Brownian motion, matrix valued L´evy process, confluent hypergeometric function, type G, semicircle law. On leave from CIMAT. This work was started while the author visited the Issac Newton Institute for Mathematical Sciences at Cambridge. He acknowledges the support of this Institute. 0031-5303/2006/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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d if there exists an ensemble of non-diagonal random matrices X d such that µ X d converges to µ in some sense when the dimension d goes to infinity. In particular, d d = E µ X we are interested in the situation where the mean spectral measure µX d d converges in distribution to µ, where E denotes expected value. In this case we refer to µ as an asymptotic mean spectral measure. The classical example is the semicircle or Wigner distribution which is the asymptotic mean spectral measure for several ensembles of random matrices, including the familiar Gaussian Unitary Ensemble (GUE). Another well known example is the Marchenko–Pastur or free Poisson law, which is the asymptotic mean spectral measure for the so called Laguerre ensemble, where each X d is a Wishart random matrix. We refer to [19], [22], [29], [42] for these and other examples and Forrester et al. [16] for a survey of recent developments in random matrix theory. In this paper we are concerned with random matrix models for distributions that are variance mixture of Gaussian laws and the relation between classical infinitely divisible properties
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