Large sample behaviour of high dimensional autocovariance matrices

TL;DR

This paper establishes the existence of limiting spectral distributions for a broad class of high-dimensional autocovariance matrices under weaker conditions, using free probability and moments methods, with applications in statistical inference.

Contribution

It provides a unified framework for LSD existence of symmetric polynomials of autocovariance matrices under weaker assumptions, extending previous results.

Findings

LSD exists for various symmetric polynomials of autocovariance matrices.

The approach uses free probability and moments for general descriptions.

Results have applications in order determination and white noise testing.

Abstract

The existence of limiting spectral distribution (LSD) of $\hat{\Gamma}_u+\hat{\Gamma}_u^*$, the symmetric sum of the sample autocovariance matrix $\hat{\Gamma}_u$ of order $u$, is known when the observations are from an infinite dimensional vector linear process with appropriate (strong) assumptions on the coefficient matrices. Under significantly weaker conditions, we prove, in a unified way, that the LSD of any symmetric polynomial in these matrices such as $\hat{\Gamma}_u+\hat{\Gamma}_u^*$, $\hat{\Gamma}_u\hat{\Gamma}_u^*$, $\hat{\Gamma}_u\hat{\Gamma}_u^*+\hat{\Gamma}_k\hat{\Gamma}_k^*$ exist. Our approach is through the more intuitive algebraic method of free probability in conjunction with the method of moments. Thus, we are able to provide a general description for the limits in terms of some freely independent variables. All the previous results follow as special cases. We suggest statistical uses of these LSD and related results in order determination and white noise testing.