Generalization of the Marčenko-Pastur problem.

Abstract

We study the spectrum of generalized Wishart matrices, defined as F=(XY^{⊤}+YX^{⊤})/2T, where X and Y are N×T matrices with zero mean, unit variance independent and identically distributed entries and such that E[X_{it}Y_{jt}]=cδ_{i,j}. The limit c=1 corresponds to the Marčenko-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.