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

الأصل · EN

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≤ n the number of variates and Xₙ,ᵣ such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of Xₙ,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 ≤ 1. We derive limit theorems for the sequence of processes with independent increments {n⁻¹ Xn, nt, t ∈ [0, T]}ₙ for T ≤ 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 β models, if we define the determinant as the product of charges.