Spiked eigenvalues of high-dimensional sample autocovariance matrices: CLT and applications

TL;DR

This paper establishes a CLT for spiked eigenvalues of high-dimensional sample autocovariance matrices, allowing for flexible divergence and applications in statistical testing and clustering of high-dimensional time series.

Contribution

It develops a general CLT for spiked eigenvalues of high-dimensional autocovariance matrices, accommodating diverging spikes and varying numbers of spikes.

Findings

CLT for spiked eigenvalues under general conditions

A novel autocovariance test for high-dimensional time series

Hierarchical clustering method applied to mortality data

Abstract

High-dimensional autocovariance matrices play an important role in dimension reduction for high-dimensional time series. In this article, we establish the central limit theorem (CLT) for spiked eigenvalues of high-dimensional sample autocovariance matrices, which are developed under general conditions. The spiked eigenvalues are allowed to go to infinity in a flexible way without restrictions in divergence order. Moreover, the number of spiked eigenvalues and the time lag of the autocovariance matrix under this study could be either fixed or tending to infinity when the dimension p and the time length T go to infinity together. As a further statistical application, a novel autocovariance test is proposed to detect the equivalence of spiked eigenvalues for two high-dimensional time series. Various simulation studies are illustrated to justify the theoretical findings. Furthermore, a hierarchical clustering approach based on the autocovariance test is constructed and applied to clustering mortality data from multiple countries.