Limit Theorems for Weakly Dependent Non-stationary Random Field Arrays and Asymptotic Inference of Dynamic Spatio-temporal Models

TL;DR

This paper establishes fundamental limit theorems for weakly dependent, non-stationary random field arrays, enabling asymptotic inference for high-dimensional dynamic spatio-temporal models including network autoregression.

Contribution

It introduces new weak dependence conditions that are preserved under transformations, facilitating the proof of LLN and CLT for complex non-stationary arrays and enabling inference in high-dimensional models.

Findings

Proved LLN and CLT for weakly dependent non-stationary arrays.

Established consistency and asymptotic normality of MLE in high-dimensional models.

Derived asymptotic properties for network autoregression models.

Abstract

We obtain the law of large numbers (LLN) and the central limit theorem (CLT) for weakly dependent non-stationary arrays of random fields with asymptotically unbounded moments. The weak dependence condition for arrays of random fields is proved to be inherited through transformation and infinite shift. This paves a way to prove the consistency and asymptotic normality of maximum likelihood estimation for dynamic spatio-temporal models (i.e. so-called ultra high-dimensional time series models) when the sample size and/or dimension go to infinity. Especially the asymptotic properties of estimation for network autoregression are obtained under reasonable regularity conditions.