2005/09/01 by Alain Rouault, Rouault, Alain
Mathematics · #15A52 (Primary) 15A15 #60F10 #60F17 #62H10 (Secondary) #FOS: Mathematics #Probability (math.PR) #math.PR #msc:15A15 #msc:15A52 #msc:60F10 #msc:60F17 #msc:62H10
paper · pdf · doi:10.48550/arxiv.math/0509021
arxiv created 2005/09/01 · arxiv updated 2009/12/01
This paper concentrates on asymptotic properties of determinants of some random symmetric matrices. If Bn,r is a n x r rectangular matrix and Bn,r' its transpose, we study det (Bn,r'Bn,r) when n,r tends to infinity with r/n → c∈ (0,1). The r column vectors of Bn,r are chosen independently, with common distribution νn. The Wishart ensemble corresponds to νn = \cal N(0, In), the standard normal distribution. We call uniform Gram ensemble the ensemble corresponding to νn = σn, the uniform distribution on the unit sphere `Sn-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 \(1)/(n)log det(Bn,\lfloor nt\rfloor'Bn,\lfloor nt\rfloor), t ∈ [0,1]\ and its asymptotic behavior as n→ ∞: a.s. convergence, fluctuations, large deviations. We connect the results for marginals (fixed t) with those obtained by the spectral method.