Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

Abstract

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from n independent observations of a p-dimensional time series with iid components converge almost surely to (1+ √ γ) 2 and (1− √ γ) 2 , respectively, as n → ∞, if p/n → γ ∈ (0, 1] and the truncated variance of the entry distribution is "almost slowly varying", a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marčenko-Pastur law, which extends a result in [7]. We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study. In our proofs we use the method of moments combined with a Path-Shortening Algorithm, which efficiently uses the structure of sample correlation matrices, to calculate precise bounds for matrix norms. We believe that this new approach could be of further use in random matrix theory.1991 Mathematics Subject Classification. Primary 60B20; Secondary 60F05 60F10 60G10 60G55 60G70.

Thin-shell theory for rotationally invariant random simplices

“…To the best of our knowledge, the most general setting in which the limiting distribution of the log-volume (or equivalently the log-determinant) was derived was in [9,60] who assumed that the entries of Y possess a finite fourth moment, which is the typical assumption in papers on linear spectral statistics. We refer to [8,18,27,28,21,11,29,30] for collections of results which show the stark differences in the asymptotic behavior under infinite fourth moments.…”

Thin-shell theory for rotationally invariant random simplices

“…To the best of our knowledge, the most general setting in which the limiting distribution of the log-volume (or equivalently the log-determinant) was derived was in [9,59] who assumed that the entries of Y possess a finite fourth moment, which is the typical assumption in papers on linear spectral statistics. We refer to [8,18,27,28,21,11,29,30] for collections of results which show the stark differences in the asymptotic behavior under infinite fourth moments.…”

Large sample correlation matrices: a comparison theorem and its applications

“…Sometimes practitioners would like to know "to which extent the random matrix results would hold if one were concerned with sample correlation matrices and not sample covariance matrices [19]". In case the elements of the data matrix X are iid with zero mean, variance equal to one and finite fourth moment it is shown by Jiang [26] (see also [19] and [23]) that the Marčenko-Pastur law is still valid for the sample correlation matrix R. The first result for the linear spectral statistics of R was proved in [21] under existence of the fourth moment. Moreover, the properly normalized largest off-diagonal entry of R converges to a Gumbel distribution as shown in [25] and recently generalized to a point process setting in [24].…”