Compound real Wishart and q-Wishart matrices

We introduce a family of matrices with non-commutative entries that generalize the classical real Wishart matrices. With the help of the Brauer product, we derive a non-asymptotic expression for the moments of traces of monomials in such matrices; the expression is quite similar to the formula derived in our previous work 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.