  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.
