Borel theorems for random matrices from the classical compact symmetric spaces

Details Borel theorems for random matrices from the classical compact symmetric spaces Borel theorems for random matrices from the classical compact symmetric spaces

2006-11-22

arxiv.org/abs/math/0611708v2

Annals of Probability 2008, Vol. 36, No. 3, 876-895

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AOP341 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/07-AOP341

We study random vectors of the form $(\operatorname {Tr}(A^{(1)}V),...,\operatorname {Tr}(A^{(r)}V))$, where $V$ is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the $A^{(\nu)}$ are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et al. for Haar distributed matrices from the classical compact groups. The proof uses integration formulas, due to Collins and \'{S}niady, for polynomial functions on the classical compact groups.

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