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High-level moving excursions for spatiotemporal Gaussian random fields with long range dependence

N. N. Leonenko, M. D. Ruiz-Medina

TL;DR

The paper develops a unified framework for the asymptotic behavior of moving-level spatiotemporal geometric functionals of long-range dependent Gaussian random fields. By leveraging a Gaussian chaos (Hermite) expansion and reduction principles under increasing-domain scaling, it derives Gaussian limit laws for a broad class of sojourn measures and Minkowski functionals, including time-varying subordination and moving thresholds $u(T)$ with $\Lambda(T)=T^{\gamma}$. The results address both separable and nonseparable covariances (notably within the Gneiting class) and extend to spherical STGRFs, yielding central limit theorems for functionals constrained to the sphere and for moving-level analyses relevant to cosmology. These theoretical findings enable robust morphometric inference and non-Gaussianity testing in applications such as Cosmic Microwave Background analysis and other spatiotemporal physical systems where structural changes evolve over time.

Abstract

The asymptotic behavior of an extended family of integral geometric random functionals, including spatiotemporal Minkowski functionals under moving levels, is analyzed in this paper. Specifically, sojourn measures of spatiotemporal long-range dependence (LRD) Gaussian random fields are considered in this analysis. The limit results derived provide general reduction principles under increasing domain asymptotics in space and time. The case of time-varying thresholds is also studied. Thus, the family of morphological measures considered allows the statistical and geometrical analysis of random physical systems displaying structural changes over time. Motivated by cosmological applications, the derived results are applied to the context of sojourn measures of spatiotemporal spherical Gaussian random fields. The results are illustrated for some families of spatiotemporal Gaussian random fields displaying complex spatiotemporal dependence structures.

High-level moving excursions for spatiotemporal Gaussian random fields with long range dependence

TL;DR

The paper develops a unified framework for the asymptotic behavior of moving-level spatiotemporal geometric functionals of long-range dependent Gaussian random fields. By leveraging a Gaussian chaos (Hermite) expansion and reduction principles under increasing-domain scaling, it derives Gaussian limit laws for a broad class of sojourn measures and Minkowski functionals, including time-varying subordination and moving thresholds with . The results address both separable and nonseparable covariances (notably within the Gneiting class) and extend to spherical STGRFs, yielding central limit theorems for functionals constrained to the sphere and for moving-level analyses relevant to cosmology. These theoretical findings enable robust morphometric inference and non-Gaussianity testing in applications such as Cosmic Microwave Background analysis and other spatiotemporal physical systems where structural changes evolve over time.

Abstract

The asymptotic behavior of an extended family of integral geometric random functionals, including spatiotemporal Minkowski functionals under moving levels, is analyzed in this paper. Specifically, sojourn measures of spatiotemporal long-range dependence (LRD) Gaussian random fields are considered in this analysis. The limit results derived provide general reduction principles under increasing domain asymptotics in space and time. The case of time-varying thresholds is also studied. Thus, the family of morphological measures considered allows the statistical and geometrical analysis of random physical systems displaying structural changes over time. Motivated by cosmological applications, the derived results are applied to the context of sojourn measures of spatiotemporal spherical Gaussian random fields. The results are illustrated for some families of spatiotemporal Gaussian random fields displaying complex spatiotemporal dependence structures.
Paper Structure (15 sections, 6 theorems, 124 equations)

This paper contains 15 sections, 6 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Connections with Statistical Physics
  3. State of the art
  4. Our contribution
  5. Preliminaries
  6. Reduction theorems for spatiotemporal random fields with LRD
  7. Separable covariance functions
  8. Sojourn functionals for spatiotemporal fields
  9. Examples
  10. Example 1
  11. Example 2
  12. Reduction theorem for time varying subordination
  13. First Minkowski functional with moving level
  14. Sojourn functionals for a class of spherical random fields
  15. Spherical spatiotemporal covariance functions

Key Result

Theorem 1

Under Conditions 1,2,3 and 4, the random variables have the same limiting distributions as $T\rightarrow \infty$ (if one of them exists).

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Theorem 2
  • Theorem 3
  • Remark 5
  • Remark 6
  • Theorem 4
  • ...and 3 more