Construction and limit theorems for supCAR fields

TL;DR

This paper introduces supCAR fields, a new class of infinitely divisible random fields with flexible dependence structures, and establishes their existence and limit theorems for statistical inference.

Contribution

It defines supCAR fields as superpositions of autoregressive fields, characterizes their properties, and derives functional limit theorems under general conditions.

Findings

Existence of supCAR fields is established.

Four types of limit theorems depending on marginals and superposition.

Explicit examples of supCAR fields with simple forms.

Abstract

The paper introduces a new class of random fields, supCAR fields, which are constructed as superpositions of continuous autoregressive random fields. These supCAR fields possess infinitely divisible marginal distributions. Their second-order properties are characterised by a novel family of covariance functions which can exhibit short- and long-range spatial dependencies. First, the existence of such fields is examined. Then, functional limit theorems for supCAR fields are derived under general assumptions. Four limiting scenarios that depend on the marginals of the underlying autoregressive fields and the specifications of the superposition are identified. Examples of specific supCAR fields, for which the assumptions and results are provided in simple, explicit forms, are presented. The obtained limit theorems can be employed for the statistical inference of supCAR fields.