LIMITING SPECTRAL DISTRIBUTION FOR LARGE SAMPLE COVARIANCE MATRICES WITH GRAPH-DEPENDENT ELEMENTS

For sample covariance matrices associated with random vectors having graph -dependent entries and a number of dimensions growing with the sample size, we derive sharp con-ditions for the limiting spectrum of the matrices to have the same form as in the case of Gaussian data with similar covariance structure. Our results are tight. In particular, they give necessary and sufficient conditions for the Marchenko-Pastur theorem for sample covariance matrices associated with random vectors having m-dependent orthonormal elements when m = o(n).