The Annals of Probability

Borel theorems for random matrices from the classical compact symmetric spaces

Benoît Collins and Michael Stolz

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We study random vectors of the form (Tr(A(1)V), …, Tr(A(r)V)), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A(ν) 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 Śniady, for polynomial functions on the classical compact groups.

Article information

Ann. Probab., Volume 36, Number 3 (2008), 876-895.

First available in Project Euclid: 9 April 2008

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Zentralblatt MATH identifier

Primary: 15A52 60F05: Central limit and other weak theorems
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 43A75: Analysis on specific compact groups

Random matrices symmetric spaces central limit theorem matrix integrals classical invariant theory


Collins, Benoît; Stolz, Michael. Borel theorems for random matrices from the classical compact symmetric spaces. Ann. Probab. 36 (2008), no. 3, 876--895. doi:10.1214/07-AOP341.

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