On a Generalisation of the Marcenko-Pastur Problem

Abstract

We study the spectrum of generalized Wishart matrices, defined as $\mathbf{F}=(X{Y}^{\top }+Y{X}^{\top })/2T$, where $X$ and $Y$ are $N\times T$ matrices with zero mean, unit variance IID entries and such that $\mathbb{E}[X_{it}{Y}_{jt}]=c{\delta }_{i,j}$. The limit $c=1$ corresponds to the Marcenko-Pastur problem. For a general $c$, we show that the Stietjes transform of $\mathbf{F}$ is the solution of a cubic equation. In the limit $c=0$, $T\gg N$ the density of eigenvalues converges to the Wigner semi-circle.

5 pages, 3 figures, reference to the "Tetilla" law added

References in corpus (2)

• Complete diagrammatics of the single ring theorem
• Optimal cleaning for singular values of cross-covariance matrices