Matrix polynomial generalizations of the sample variance-covariance matrix when pn⁻¹ → y ∈ (0, ∞)

Abstract

Let {Z _u = ((ε_{u, i, j}))_p×n} be random matrices where {ε_{u, i, j}} are independently distributed. Suppose {A _i }, {B _i } are non-random matrices of order p × p and n × n respectively. Consider all p × p random matrix polynomials $P = \prod\nolimits_{i = 1}^{k_l } {\left( {n^{ - 1} A_{t_i } Z_{j_i } B_{s_i } Z_{j_i }^* } \right)A_{t_{k_l + 1} } }$. We show that under appropriate conditions on the above matrices, the elements of the non-commutative *-probability space Span {P} with state p⁻¹ETr converge. As a by-product, we also show that the limiting spectral distribution of any self-adjoint polynomial in Span{P} exists almost surely.