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

Johannes Heiny and Thomas Mikosch

Stochastic Processes and their Applications, 2018, vol. 128, issue 8, 2779-2815

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 Bai and Zhou (2008). 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.

Keywords: Sample correlation matrix; Infinite fourth moment; Largest eigenvalue; Smallest eigenvalue; Spectral distribution; Sample covariance matrix; Self-normalization; Regular variation; Combinatorics (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:128:y:2018:i:8:p:2779-2815

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DOI: 10.1016/j.spa.2017.10.002

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