Large sample correlation matrices: a comparison theorem and its applications

In this paper, we show that the diagonal of a high-dimensional sample covariance matrix stemming from n independent observations p-dimensional time series with finite fourth moments can be approximated in spectral norm by population matrix. We assume n,p→∞ p∕n tending to constant which might positive or zero. As applications, provide an approximation correlation R and derive variety results for its eigenvalues. identify limiting distribution construct estimator Finally, almost sure limits extreme eigenvalues generalized spiked model are analyzed.