Generalization Of The Marcenko-Pastur Problem

We study the spectrum of generalized Wishart matrices, defined as F = (XY inverted perpendicular YX inverted perpendicular)/2T, where X and Y are N x T matrices with zero mean, unit variance independent and identically distributed entries and such that E[XitYit] = c delta(i,j). The limit c = 1 corresponds to the Marcenko-Pastur problem. For a general c, we show that the Stieltjes transform of F is the solution of a cubic equation. In the limit c = 0, T >> N, the density of eigenvalues converges to the Wigner semicircle.