Limiting spectral distribution of sample autocovariance matrices

TL;DR

This paper investigates the asymptotic behavior of the spectral distribution of sample autocovariance matrices for linear processes, revealing conditions under which the empirical spectral distribution converges and how it relates to the theoretical distribution.

Contribution

It establishes the convergence of the empirical spectral distribution of sample autocovariance matrices for linear processes and characterizes the limits under various matrix modifications and banding conditions.

Findings

The ESD converges as dimension increases for linear processes.

The limit distribution is independent of the underlying i.i.d. sequence's distribution.

Banded matrices have unbounded support unless the bandwidth ratio tends to zero.

Abstract

We show that the empirical spectral distribution (ESD) of the sample autocovariance matrix (ACVM) converges as the dimension increases, when the time series is a linear process with reasonable restriction on the coefficients. The limit does not depend on the distribution of the underlying driving i.i.d. sequence and its support is unbounded. This limit does not coincide with the spectral distribution of the theoretical ACVM. However, it does so if we consider a suitably tapered version of the sample ACVM. For banded sample ACVM the limit has unbounded support as long as the number of non-zero diagonals in proportion to the dimension of the matrix is bounded away from zero. If this ratio tends to zero, then the limit exists and again coincides with the spectral distribution of the theoretical ACVM. Finally, we also study the LSD of a naturally modified version of the ACVM which is not non-negative definite.