Borel theorems for random matrices from the classical compact symmetric spaces

• 成果类型: Article
• 署名作者: Collins, Benoit; Stolz, Michael
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; University of Ottawa; Ruhr University Bochum
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP341
• 发表日期: 2008
• 页码: 876-895
• 关键词: eigenvalues; ensembles

摘要

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 AM 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 Sniady, for polynomial functions on the classical compact groups.