A random matrix model for the Gaussian distribution

SummaryThe 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 <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>2$-gamma distribution.

[1]  F. Hiai,et al.  The semicircle law, free random variables, and entropy , 2006 .

[2]  F. Steutel,et al.  Infinite Divisibility of Probability Distributions on the Real Line , 2003 .

[3]  STABLE LAWS AND DOMAINS OF ATTRACTION IN FREE PROBABILITY THEORY , 1999, math/9905206.

[4]  Lucy Joan Slater.  Confluent Hypergeometric Functions , 1960 .

[5]  O. Barndorff-Nielsen,et al.  Multivariate subordination, self-decomposability and stability , 2001, Advances in Applied Probability.

[6]  Marc Yor,et al.  Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning , 2003 .

[7]  Michel Ledoux,et al.  Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case , 2004 .

[8]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[9]  D. Pollard.  Convergence of stochastic processes , 1984 .

[10]  Douglas Kelker,et al.  Infinite Divisibility and Variance Mixtures of the Normal Distribution , 1971 .

[11]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[12]  Lennart Bondesson,et al.  Generalized Gamma Convolutions and Related Classes of Distributions and Densities , 1992 .

[13]  O. Barndorff-Nielsen,et al.  A CONNECTION BETWEEN FREE AND CLASSICAL INFINITE DIVISIBILITY , 2004 .

[14]  Alexander Soshnikov,et al.  Poisson Statistics for the Largest Eigenvalues of Wigner Random Matrices with Heavy Tails , 2004 .

[15]  The arithmetical character of the Wishart distribution , 1948 .

[16]  Boris A. Khoruzhenko,et al.  Asymptotic properties of large random matrices with independent entries , 1996 .

[17]  Alexander Tchernitser,et al.  Multifactor Stochastic Variance Models in Risk Management: Maximum Entropy Approach and Levy Processes , 2001 .

[18]  P. J. Forrester,et al.  Developments in random matrix theory , 2003, cond-mat/0303207.

[19]  Ken-iti Sato.  Subordination and self-decomposability , 2001 .

[20]  R. Rao,et al.  Normal Approximation and Asymptotic Expansions , 1976 .

[21]  S. Thorbjørnsen,et al.  Random matrices with complex Gaussian entries , 2003 .

[22]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[23]  M. Marcus.  ξ-radial processes and random Fourier series , 1987 .

[24]  C. Halgreen.  Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions , 1979 .