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.