On the smallest eigenvalues of covariance matrices of multivariate spatial processes

TL;DR

This paper investigates the asymptotic behavior of the smallest eigenvalues of covariance matrices in multivariate spatial processes, establishing conditions under which they remain positive definite as the domain grows.

Contribution

It provides theoretical results on the positive definiteness of covariance matrices in multivariate spatial models under weak assumptions, filling a gap in asymptotic analysis.

Findings

Smallest eigenvalue remains bounded away from zero asymptotically.

Weak assumptions on covariance functions are sufficient for positive definiteness.

Implications for parameter estimation and model validity in spatial statistics.

Abstract

There has been a growing interest in providing models for multivariate spatial processes. A majority of these models specify a parametric matrix covariance function. Based on observations, the parameters are estimated by maximum likelihood or variants thereof. While the asymptotic properties of maximum likelihood estimators for univariate spatial processes have been analyzed in detail, maximum likelihood estimators for multivariate spatial processes have not received their deserved attention yet. In this article we consider the classical increasing-domain asymptotic setting restricting the minimum distance between the locations. Then, one of the main components to be studied from a theoretical point of view is the asymptotic positive definiteness of the underlying covariance matrix. Based on very weak assumptions on the matrix covariance function we show that the smallest eigenvalue of the covariance matrix is asymptotically bounded away from zero. Several practical implications are discussed as well.