Large sample correlation matrices: a comparison theorem and its applications

2022/01/04 by Heiny, Johannes · 1 citation
#FOS: Mathematics #Primary 60B20 #Probability (math.PR) #Secondary 60F05 60G10 60G57 60G70 #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2201.00916

Abstract

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 \mathbf R and derive a variety of results for its eigenvalues. We identify the limiting spectral distribution of \mathbf R and construct an estimator for the population correlation matrix and its eigenvalues. Finally, the almost sure limits of the extreme eigenvalues of \mathbf R in a generalized spiked correlation model are analyzed.

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