Some remarks on the Dozier-Silverstein theorem for random matrices with dependent entries (Q2844433)

Consider a sequence of deterministic \(n\times N_n\)-matrices \(R_n\) such that \(\lim_{n\to\infty} n/N_n=c\in (0,\infty)\) and assume that the spectral distribution of the matrix \(\frac 1{N_n} R_n R_n^*\) converges weakly to some probability measure on \(\mathbb R\) with distribution function \(H\), as \(n\to\infty\). The paper studies information plus noise matrices of the form NEWLINE\[NEWLINE M_n = \frac 1 {N_n} (R_n + X_n) (R_n + X_n)^*, NEWLINE\]NEWLINE where \(X_n\) is some random \(n\times N_n\)-matrix. A theorem of \textit{R.\ B.\ Dozier} and \textit{J.\ W.\ Silverstein} [J. Multivariate Anal. 98, No. 6, 1099--1122 (2007; Zbl 1127.62015)] states that if the entries of \(X_n\) are i.i.d.\ mean zero, variance \(1\) random variables, then the spectral distribution of \(M_n\) converges weakly to some probability measure on \(\mathbb R\) characterized in terms of \(H\). In the present paper, the author shows that modifying the original argument of Dozier and Silverstein it is possible to extend their result to more general noise matrices \(X_n\). In particular, the author considers matrices with independent columns satisfying certain concentration inequality for quadratic forms, matrices with independent entries satisfying a Lindeberg-type condition and certain classes of matrices with dependencies among columns. The author proves a version of the Dozier-Silverstein result for matrices with i.i.d.\ entries belonging to the domain of attraction of the normal law (but having infinite variance). As a corollary of his results, the author proves a circular law for random \(n\times n\)-matrices whose columns are independent identically distributed random vectors sampled from an isotropic log-concave distribution on \(\mathbb R^n\).