Polynomial generalizations of the sample variance-covariance matrix when pn−1 → 0

Abstract

Let [Formula: see text] be random matrices, where [Formula: see text] are independently distributed. Suppose [Formula: see text], [Formula: see text] are non-random matrices of order [Formula: see text] and [Formula: see text] respectively. Suppose [Formula: see text], [Formula: see text] and [Formula: see text]. Consider all [Formula: see text] random matrix polynomials constructed from the above matrices of the form [Formula: see text] [Formula: see text] and the corresponding centering polynomials [Formula: see text] [Formula: see text]. We show that under appropriate conditions on the above matrices, the variables in the non-commutative ∗-probability space [Formula: see text] with state [Formula: see text] converge. We also show that the limiting spectral distribution of [Formula: see text] exists almost surely whenever [Formula: see text] and [Formula: see text] are self-adjoint. The limit can be expressed in terms of, semi-circular, circular and other families and, limits of [Formula: see text], [Formula: see text] and non-commutative limit of [Formula: see text]. Our results fully generalize the results already known for [Formula: see text].