The convergence on spectrum of sample covariance matrices for information-plus-noise type data

In this paper, we consider the limiting spectral distribution of the information-plusnoise type sample covariance matrices C _n = 1/N (R _n +??X _n )(R _n + ??X _n )*, under the assumption that the entries of X _n are independent but non-identically distributed random variables. It is proved that, almost surely, the empirical spectral distribution of C _n converges weakly to a nonrandom distribution whose Stieltjes transform satisfies a certain equation. Our result extends the previous one with the entries of X _n are i.i.d. random varibles to a more general case. The proof of the result mainly employs the Stein equation and the cumulant expansion formula of independent random variables.