Limiting eigenvalue behavior of a class of large dimensional random matrices formed from a Hadamard product

Abstract

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices [Formula: see text], studied in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. [Formula: see text] (Kluwer Academic Publishers, Dordrecht, 2001)]. Here, [Formula: see text] is an [Formula: see text] random matrix consisting of independent complex standardized random variables, [Formula: see text], [Formula: see text], has nonnegative entries, and ∘ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of [Formula: see text] and [Formula: see text] which are different from those in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. 1 (Kluwer Academic Publishers, Dordrecht, 2001)], which include a Lindeberg condition on the entries of [Formula: see text], as well as a bound on the average of the rows and columns of [Formula: see text]. The present paper separates the assumptions needed on [Formula: see text] and [Formula: see text]. It assumes a Lindeberg condition on the entries of [Formula: see text], along with a tightness-like condition on the entries of [Formula: see text].