Eigenvalue distribution of a high-dimensional distance covariance matrix with application

TL;DR

This paper studies the eigenvalue distribution of a new high-dimensional distance covariance matrix, deriving its limit behavior and phase transition phenomena, with applications in detecting weak dependence.

Contribution

It introduces a novel random matrix model for distance covariance, deriving its eigenvalue distribution limit and identifying phase transition behavior for dependence detection.

Findings

Eigenvalue distribution converges to a deterministic limit in high dimensions.

Top eigenvalues exhibit a phase transition under finite-rank dependence.

New method for detecting weak dependence in high-dimensional data.

Abstract

We introduce a new random matrix model called distance covariance matrix in this paper, whose normalized trace is equivalent to the distance covariance. We first derive a deterministic limit for the eigenvalue distribution of the distance covariance matrix when the dimensions of the vectors and the sample size tend to infinity simultaneously. This limit is valid when the vectors are independent or weakly dependent through a finite-rank perturbation. It is also universal and independent of the details of the distributions of the vectors. Furthermore, the top eigenvalues of this distance covariance matrix are shown to obey an exact phase transition when the dependence of the vectors is of finite rank. This finding enables the construction of a new detector for such weak dependence where classical methods based on large sample covariance matrices or sample canonical correlations may fail in the considered high-dimensional framework.