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

TY - JOUR

T1 - Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

AU - Heiny, Johannes

AU - Mikosch, Thomas Valentin

PY - 2018/8

Y1 - 2018/8

N2 - 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. 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.

AB - 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. 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.

KW - Combinatorics

KW - Infinite fourth moment

KW - Largest eigenvalue

KW - Primary

KW - Regular variation

KW - Sample correlation matrix

KW - Sample covariance matrix

KW - Secondary

KW - Self-normalization

KW - Smallest eigenvalue

KW - Spectral distribution

U2 - 10.1016/j.spa.2017.10.002

DO - 10.1016/j.spa.2017.10.002

M3 - Journal article

AN - SCOPUS:85032944975

SN - 0304-4149

VL - 128

SP - 2779

EP - 2815

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

IS - 8

ER -