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Article Dans Une Revue ALEA : Latin American Journal of Probability and Mathematical Statistics Année : 2007

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

Alain Rouault

Résumé

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.

Dates et versions

hal-00008371 , version 1 (02-09-2005)

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Alain Rouault. Pathwise asymptotic behavior of random determinants in the uniform Gram and Wishart ensembles. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2007, 3, pp.181-230. ⟨hal-00008371⟩
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