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

TL;DR

This paper investigates the asymptotic behavior of determinants of random matrices from Laguerre, Gram, and Jacobi ensembles, deriving limit theorems for their logarithms as matrix size and variate count grow, with connections to spectral distribution methods.

Contribution

It generalizes Bartlett-type theorems to decompose determinants into independent gamma or beta variables and studies their asymptotic processes as dimensions increase.

Findings

Limit theorems for processes with independent increments of log-determinants.

Connections established between spectral distribution and determinant asymptotics.

Results applicable to beta models via charges.

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, $r\leq n$ 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 $r/n = t \leq 1$. 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 $T \leq 1$.. 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.