On the Asymptotic Distribution of Singular Values of Powers of Random Matrices

Abstract

We consider powers of random matrices with independent entries. Let Xij, i, j ≥ 1, be independent complex random variables with E Xij = 0 and E{pipe}Xij{pipe}2 = 1, and let X denote an n × n matrix with {pipe}X{pipe}ij - Xij for 1 ≤ i, j ≤ n. Denote by s1(m)≥ ... ≥ sn(m) the singular values of the random matrix W:= n-m/2Xm and define the empirical distribution of the squared singular values by (Formula presented.), where I{B} denotes the indicator of an event B. We prove that the expected spectral distribution Fn(m)(x) = EFn(m)(x) converges under the Lindeberg condition to the distribution function G(m)(x) defined by its moments (Formula presented.). © 2014 Springer Science+Business Media New York.