On the behaviour of large empirical autocovariance matrices between the past and the future

TL;DR

This paper analyzes the asymptotic distribution of squared singular values of sample autocovariance matrices in high-dimensional Gaussian sequences, revealing a deterministic measure and the almost sure localization of singular values.

Contribution

It characterizes the limiting distribution and support of singular values for high-dimensional autocovariance matrices using Gaussian tools, providing new theoretical insights.

Findings

Distribution converges to a deterministic measure

Singular values are almost surely near the support S

Support S is explicitly characterized

Abstract

The asymptotic behaviour of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established the distribution behaves as a deterministic probability measure whose support S is characterized. It is also established that the singular values to the square are almost surely located in a neighbourhood of S.