Pathwise asymptotic behavior of random determinants in the uniform Gram and Wishart ensembles
| dc.creator | Rouault, Alain | |
| dc.date | 2005-09-01 | |
| dc.date.accessioned | 2026-07-07T05:22:53Z | |
| dc.date.available | 2026-07-07T05:22:53Z | |
| dc.description | 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 ν_n. The Wishart ensemble corresponds to ν_n = {\cal N}(0, I_n), the standard normal distribution. We call uniform Gram ensemble the ensemble corresponding to ν_n = σ_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. | |
| dc.identifier | https://arxiv.org/abs/math/0509021 | |
| dc.identifier | http://arxiv.org/abs/math/0509021 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76233 | |
| dc.subject | Probability | |
| dc.subject | 15A52 (Primary) 15A15; 60F10; 60F17; 62H10 (Secondary) | |
| dc.title | Pathwise asymptotic behavior of random determinants in the uniform Gram and Wishart ensembles | |
| dc.type | text |