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Compound real Wishart and qWishart matrices
, 2007
"... 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 deriv ..."
Abstract

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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] for independent complex Wishart matrices. We then analyze the fluctuations about the MarchenkoPastur law. We show that after centering by the mean, traces of real symmetric polynomials in qWishart 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.