Compound Real Wishart and q-Wishart Matrices

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.