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 -