Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix

Let B n = A n + X n T n X n T , where A n is a random symmetric matrix, T n a random symmetric matrix, and X n = 1 n ( X i j ( n ) ) n × p with X i j ( n ) being independent real random variables. Suppose that X n , T n and A n are independent. It is proved that the empirical spectral distribution of the eigenvalues of random symmetric matrices B n converges almost surely to a non-random distribution.