ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES

Let x be a complex random variable such that Ex = 0, E|x|(2) = 1, and E|x|(4) < infinity. Let x(ij), i, j is an element of {1, 2,...}, be independent copies of x. Let X = ( N-(1/2) xij), 1 <= i, j <= N, be a random matrix. Writing X* for the adjoint matrix of X, consider the product (XX)-X-m*(m) with some m <= {1, 2,...}. The matrix (XX)-X-m*(m) is Hermitian positive semidefinite. Let lambda(1), lambda(2),...,lambda(N) be eigenvalues of (XX)-X-m*(m) (or squared singular values of the matrix X-m). In this paper, we find the asymptotic distribution function G((m))(x) = limN ->infinity EFN(m)(x) of the empirical distribution function F-N((m))(x) = N-1 Sigma(N)(k=1) I{lambda(k) <= x}, where I {A} stands for the indicator function of an event A. With m = 1, our result turns to a well- known result of Marchenko and Pastur [V. Marchenko and L. Pastur, The eigenvalue distribution in some ensembles of random matrices, Math. USSR Sb., 1:457- 483, 1967].