On a Generalisation of the Marcenko-Pastur Problem - Bouchaud, Jean-Philippe; Potters, Marc (arXiv.org, 2020)
We study the spectrum of generalized Wishart matrices, defined as \(F=( X Y^\top + Y X^\top)/2T\), where \(X\) and \(Y\) are \(N × T\) matrices with zero mean, unit variance IID entries and such that \(\mathbb{E}[X_{it} Y_{jt}]=c δ_{i,j}\). The limit \(c=1\) corresponds to the Marcenko-Pastur problem. For a general \(c\), we show that the Stietjes transform of \(F\) 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.