Pathwise asymptotic behavior of random determinants in the uniform Gram and Wishart ensembles

TL;DR

This paper investigates the asymptotic behavior of determinants of certain random matrices from the Wishart and uniform Gram ensembles, revealing convergence, fluctuations, and large deviations as matrix dimensions grow large.

Contribution

It extends Bartlett's theorem to the uniform Gram ensemble and analyzes the asymptotic properties of the log-determinant process for large matrices.

Findings

Almost sure convergence of the log-determinant process

Characterization of fluctuations around the limit

Large deviation principles established

Abstract

This paper concentrates on asymptotic properties of determinants of some random symmetric matrices. If B_{n,r} is a n x r rectangular matrix and B_{n,r}' its transpose, we study det (B_{n,r}'B_{n,r}) when n,r tends to infinity with r/n \to c\in (0,1). The r column vectors of B_{n,r} are chosen independently, with common distribution \nu_n. The Wishart ensemble corresponds to \nu_n = {\cal N}(0, I_n), the standard normal distribution. We call uniform Gram ensemble the ensemble corresponding to \nu_n = \sigma_n, the uniform distribution on the unit sphere `S_{n-1}. In the Wishart ensemble, a well known Bartlett's theorem decomposes the above determinant into a product of chi-square variables. The same holds in the uniform Gram ensemble. This allows us to study the process \{\frac{1}{n}\log \det\big(B_{n,\lfloor nt\rfloor}'B_{n,\lfloor nt\rfloor}\big), t \in [0,1]\} and its asymptotic behavior as n\to \infty: a.s. convergence, fluctuations, large deviations. We connect the results for marginals (fixed t) with those obtained by the spectral method.