Strong convergence of the empirical spectral distribution of a unified matrix model

Let [Formula: see text] is a positive definite matrix and [Formula: see text] with [Formula: see text] being independent and identically distributed (i.i.d.) centered real random variables. Consider the matrix [Formula: see text] where [Formula: see text] and [Formula: see text] are two projection matrices (deterministic or random) satisfying [Formula: see text] and [Formula: see text]. Additionally, if [Formula: see text] and [Formula: see text] are random matrices, they are independent of [Formula: see text]. In this paper, we demonstrate that the empirical spectral distribution of [Formula: see text] converges almost surely to a non-random distribution when [Formula: see text] and [Formula: see text], assuming [Formula: see text] has finite second moment.