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

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dc.creatorRouault, Alain
dc.date2006-07-29
dc.date2007-10-17
dc.date.accessioned2026-07-07T08:57:10Z
dc.date.available2026-07-07T08:57:10Z
dc.descriptionWe 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 $β$ models, if we define the determinant as the product of charges.
dc.description51 pages ; it replaces and extends arXiv:math/0607767 and arXiv:math/0509021 Third version: corrected constants in Theorem 3.1
dc.identifierhttps://arxiv.org/abs/math/0607767
dc.identifierhttp://arxiv.org/abs/math/0607767
dc.identifierLatin American Journal of Probability and Mathematical Statistics (ALEA) 3 (2007) 181-230
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/146868
dc.subjectProbability
dc.subject15 A 52, 15 A 15, 60F 10, 60F 17, 62 H 10
dc.titleAsymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles
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