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

Let Bn=An+XnTnXnT, where An is a random symmetric matrix, Tn a random symmetric matrix, and Xn=1n(Xij(n))n×p with Xij(n) being independent real random variables. Suppose that Xn, Tn and An are independent. It is proved that the empirical spectral distribution of the eigenvalues of random symmetric matrices Bn converges almost surely to a non-random distribution.