Singular vector distribution of sample covariance matrices

Abstract

<jats:title>Abstract</jats:title><jats:p>We consider a class of sample covariance matrices of the form <jats:italic>Q</jats:italic> = <jats:italic>TXX</jats:italic>*<jats:italic>T</jats:italic>*, where <jats:italic>X</jats:italic> = (<jats:italic>x</jats:italic><jats:sub><jats:italic>ij</jats:italic></jats:sub>) is an <jats:italic>M</jats:italic>×<jats:italic>N</jats:italic> rectangular matrix consisting of independent and identically distributed entries, and <jats:italic>T</jats:italic> is a deterministic matrix such that <jats:italic>T</jats:italic>*<jats:italic>T</jats:italic> is diagonal. Assuming that <jats:italic>M</jats:italic> is comparable to <jats:italic>N</jats:italic>, we prove that the distribution of the components of the right singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of <jats:italic>x</jats:italic><jats:sub><jats:italic>ij</jats:italic></jats:sub> coincide with the Gaussian random variables. For the right singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of <jats:italic>x</jats:italic><jats:sub><jats:italic>ij</jats:italic></jats:sub> match those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume that <jats:italic>T</jats:italic> is diagonal.</jats:p>