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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn079, 42 pages, doi:10.1093/imrn/rnn079 published on July 21, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Compound Real Wishart and q-Wishart Matrices

Wlodzimierz Bryc

Department of Mathematical Sciences, University of Cincinnati, 2855 Campus Way, PO Box 210025, Cincinnati, OH 45221-0025, USA

Correspondence: Correspondence to be sent to: brycw{at}math.uc.edu

We introduce a family of matrices with noncommutative entries that generalize the classical real Wishart matrices. With the help of the Brauer product, we derive a nonasymptotic expression for the moments of traces of monomials in such matrices; the expression is quite similar to the formula derived in [9, Theorem 2.1; Asymptotic normality for traces of polynomials in independent complex Wishart matrices; Probability Theory and Related fields, 140, 2008] for independent complex Wishart matrices.

We then analyze the fluctuations about the Marchenko–Pastur law. We show that after centering by the mean, traces of real symmetric polynomials in q-Wishart matrices converge in distribution, and we identify the asymptotic law as the normal law when q = 1, and as the semicircle law when q = 0.


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