A study on the spectral properties of covariance type matrices with isotropic log-concave column vectors

Abstract

The limiting spectral distribution of matrix B n = 1 N X n X n * T n is considered in this paper. Existing results always focus on the condition of modifying Tn, but for Xn, it is usually assumed to be a matrix composed of n × N independent identically distributed elements. Here we specify the joint distribution of column vectors of Xn. In particular, entries on the same column of Xn are correlated, in contrast with more common independence assumptions. Assuming that the columns of Xn are random vectors following the isotropic log-concave distribution, and under some additional regularity conditions, we prove that the empirical spectral distribution F B n of matrix Bn converges to a deterministic probability distribution F almost surely. Moreover, the Stieltjes transformation m = m(z) of F satisfies a deterministic form of equation, and for any z ∈ C +, it is the unique solution of the equation.