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