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

Abstract

We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If $n$ is the size of the sample, $r\le n$ the number of variates and $X_{n,r}$ such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of $\det X_{n,r}$ into a product of $r$ independent gamma or beta random variables. For $n$ fixed, we study the evolution as $r$ grows, and then take the limit of large $r$ and $n$ with $r/n=t\le 1$. We derive limit theorems for the sequence of {\it processes with independent increments} $\{n^{-1}\log \det X_{n,\lfloor nt\rfloor },t\in [0,T]{\}}_n$ for $T\le 1$.. Since the logarithm of the determinant is a linear statistic of the empirical spectral distribution, we connect the results for marginals (fixed $t$) with those obtained by the spectral method. Actually, all the results hold true for $\beta$ models, if we define the determinant as the product of charges.

51 pages ; it replaces and extends arXiv:math/0607767 and arXiv:math/0509021 Third version: corrected constants in Theorem 3.1

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• On a class of free Levy laws related to a regression problem
• Concentration of permanent estimators for certain large matrices
• Large deviations for functions of two random projection matrices
• Approximation Operators, Exponential, q-Exponential, and Free Exponential Families

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