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

Let {Z u = ((εu, i, j))p×n} be random matrices where {εu, 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 $$P = \prod\nolimits_{i = 1}^{k_l } {\left( {n^{ - 1} A_{t_i } Z_{j_i } B_{s_i } Z_{j_i }^* } \right)A_{t_{k_l + 1} } }$$ P = ∏ i = 1 k l ( n − 1 A t i Z j i B s i Z j i ∗ ) 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.