Compound real Wishart and q-Wishart matrices

TL;DR

This paper introduces a family of non-commutative matrices generalizing real Wishart matrices, derives moment formulas, and analyzes their fluctuations, revealing convergence to normal or semicircle laws depending on a parameter q.

Contribution

It extends classical Wishart matrices to a non-commutative setting and provides explicit moment formulas and fluctuation analysis for these generalized matrices.

Findings

Derived non-asymptotic moment expressions for traces of monomials.

Established convergence of traces to normal or semicircle laws depending on q.

Analyzed fluctuations around the Marchenko-Pastur law.

Abstract

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.