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Construction and limit theorems for supCAR fields

Illia Donhauzer, Nikolai Leonenko, Andriy Olenko

TL;DR

This work introduces supCAR fields, a broad class of ambit-type random fields formed as superpositions of CAR$(1)$ kernels, yielding infinitely divisible marginals and a rich covariance family that supports both short- and long-range dependence. The authors provide existence conditions, explicit marginal and joint cumulant representations, and covariance/spectral-density formulae showing the dependence structure is governed by the mixing measure $\pi$, independent of CAR$(1)$ marginals. They establish functional limit theorems for the integrated field $X^*(T)$ across four regimes determined by tail indices and the presence of a Gaussian component, yielding Brownian, generalized Brownian, and stable process limits. These results enable principled statistical inference for supCAR fields and offer a flexible framework for modeling spatial data with diverse local and asymptotic behaviors.

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.

Construction and limit theorems for supCAR fields

TL;DR

This work introduces supCAR fields, a broad class of ambit-type random fields formed as superpositions of CAR kernels, yielding infinitely divisible marginals and a rich covariance family that supports both short- and long-range dependence. The authors provide existence conditions, explicit marginal and joint cumulant representations, and covariance/spectral-density formulae showing the dependence structure is governed by the mixing measure , independent of CAR marginals. They establish functional limit theorems for the integrated field across four regimes determined by tail indices and the presence of a Gaussian component, yielding Brownian, generalized Brownian, and stable process limits. These results enable principled statistical inference for supCAR fields and offer a flexible framework for modeling spatial data with diverse local and asymptotic behaviors.

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.

Paper Structure

This paper contains 5 sections, 11 theorems, 256 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Superpositions of continuous autoregressive fields
  4. Limit theorems for supCAR fields
  5. Conclusions and future work

Key Result

Theorem 2.1

brockwell2017continuous Let $L(\cdot)$ be the second-order Lévy sheet with parameters $\mu$ and $\sigma^2$ defined in musigma. Then, the isotropic random field is called CAR(1) field. Its mean equals the spectral density of $S_\lambda(\cdot)$ is and the covariance function of $S_\lambda(\cdot)$ is given by where and $K_{d/2+1}(\cdot)$ denotes the modified Bessel function of the second kind o

Figures (4)

  • Figure 1: Realisations of CAR(1) fields $S_\lambda(\cdot)$ with marginal Gamma distributions
  • Figure 2: Realisations of $X(\cdot)$ with marginal Gamma distributions and $\pi(\cdot)$ given by \ref{['gamma:density']}
  • Figure 3: Normalised covariances of CAR (dotted lines) and supCAR (solid lines) fields
  • Figure 4: Parameter space diagram of possible limit scenarios

Theorems & Definitions (30)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • proof
  • Corollary 3.4
  • Example 3.5
  • Proposition 3.6
  • ...and 20 more