Limiting Spectral Distribution for Large Sample Covariance Matrices with Graph-Dependent Elements

We obtain the limiting spectral distribution for large sample covariance matrices associated with random vectors having graph-dependent entries under the assumption that the interdependence among the entries grows with the sample size n. Our results are tight. In particular, they give necessary and sufficient conditions for the Marchenko-Pastur theorem for sample covariance matrices with m-dependent orthonormal elements when m = o(n).