Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles

By Alain Rouault. 2006

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We consider properties of determinants of some random symmetric ma- trices issued from multivariate statistics: Wishart/Laguerre ensemble (sample co- variance matrices), Uniform Gram ensemble (sample correlation matrices) and Ja- cobi ensemble (MANOVA). If n is the size of the sample, r n the number of variates and Xn;r such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of detXn;r into a product of r independent Gamma or Beta random variables. For n xed, we study the evolution as r grows, and then take the limit of large r and n with r=n = t 1. We derive limit theorems for the sequence of processes with independent increments fn 1 log detXn;bntc;t 2 (0;T )gn for T 1: convergence in probability, invariance principle, large deviations. Since the logarithm of the determinant is a linear statistic of the empirical spectral dis- tribution, we connect the results for marginals (xed t) with those obtained by the spectral method. Actually, all the results hold true for Coulomb gases or -models, if we dene the determinant as the product of charges. The classical matrix models (real, complex, and quaternionic) correspond to the particular values = 1; 2; 4 of the Dyson parameter.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/507f31c6d649d31608e08e78032f555fa895dea0