Journal of Theoretical Probability

, Volume 24, Issue 4, pp 1044–1062

Genus Expansion for Real Wishart Matrices

Article

DOI: 10.1007/s10959-010-0278-7

Cite this article as:
Redelmeier, C.E.I. J Theor Probab (2011) 24: 1044. doi:10.1007/s10959-010-0278-7

Abstract

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central-limit-type theorem for the asymptotic distribution around the expected value.

Keywords

Real Wishart matrices Genus expansion Maps on non-orientable surfaces Central limit theorem 

Mathematics Subject Classification (2000)

15A52 60G15 05A15 

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsQueen’s UniversityKingstonCanada

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