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Noncentral limit results for spatiotemporal random fields on manifolds and beyond

M. D. Ruiz-Medina

TL;DR

The paper addresses noncentral limit behavior for scaled functionals of long-range dependent Gaussian spatiotemporal random fields with Hermite rank $m_0=2$, defined on manifolds and convex domains. It develops a framework combining spatiotemporal Karhunen–Loève expansions, Fredholm determinant techniques, and reduction theorems to obtain Rosenblatt-type limit laws, expressed as infinite sums of chi-squared components across spectral eigenspaces. The results are established via characteristic-function analysis and mean-square convergence arguments in two geometric settings: sphere-like manifolds and compact convex sets with positive Lebesgue measure, with explicit representations for the limit in terms of spectral data and double Wiener–Itô integrals. The work broadens noncentral limit theory for STRFs to spatiotemporal domains on manifolds and beyond, providing precise asymptotics for nonlinear functionals of LRD Gaussian STRFs and pointing to extensions to Laguerre Chaos cases.

Abstract

This paper derives noncentral limit results (NCLTs) for suitable scaling of functionals of spatially homogeneous and isotropic, and stationary in time, LRD Gaussian subordinated Spatiotemporal Random Fields (STRFs) with Hermite rank equal to two. The cases of connected and compact two point homogeneous spaces M_{d} in R^{d+1}, and compact convex sets K in R^{d+1},$ whose interior has positive Lebesgue measure, are analyzed. These NCLTs are obtained in the second Wiener Chaos by applying reduction theorems. The methodological approaches adopted in the derivation of these results are based on the pure point and continuous spectra of the Gaussian STRFs subordinators defined on M_{d} and K, respectively.

Noncentral limit results for spatiotemporal random fields on manifolds and beyond

TL;DR

The paper addresses noncentral limit behavior for scaled functionals of long-range dependent Gaussian spatiotemporal random fields with Hermite rank , defined on manifolds and convex domains. It develops a framework combining spatiotemporal Karhunen–Loève expansions, Fredholm determinant techniques, and reduction theorems to obtain Rosenblatt-type limit laws, expressed as infinite sums of chi-squared components across spectral eigenspaces. The results are established via characteristic-function analysis and mean-square convergence arguments in two geometric settings: sphere-like manifolds and compact convex sets with positive Lebesgue measure, with explicit representations for the limit in terms of spectral data and double Wiener–Itô integrals. The work broadens noncentral limit theory for STRFs to spatiotemporal domains on manifolds and beyond, providing precise asymptotics for nonlinear functionals of LRD Gaussian STRFs and pointing to extensions to Laguerre Chaos cases.

Abstract

This paper derives noncentral limit results (NCLTs) for suitable scaling of functionals of spatially homogeneous and isotropic, and stationary in time, LRD Gaussian subordinated Spatiotemporal Random Fields (STRFs) with Hermite rank equal to two. The cases of connected and compact two point homogeneous spaces M_{d} in R^{d+1}, and compact convex sets K in R^{d+1},$ whose interior has positive Lebesgue measure, are analyzed. These NCLTs are obtained in the second Wiener Chaos by applying reduction theorems. The methodological approaches adopted in the derivation of these results are based on the pure point and continuous spectra of the Gaussian STRFs subordinators defined on M_{d} and K, respectively.
Paper Structure (7 sections, 3 theorems, 46 equations)

This paper contains 7 sections, 3 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. NCLT via characteristic function
  4. NCLT in the pure point spectral domain
  5. Mean--square convergence for STRFs on compact sets of positive Lebesgue measure
  6. Mean--square convergence for STRFs on $\mathbb{S}_{d}(1)$
  7. Final Comments

Key Result

Theorem 1

Under Assumptions A0--A2, the following weak convergence holds: where and the limit random variable $S_{\infty}$ has characteristic function given by with, for $m\geq 2,$

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Definition 1
  • Remark 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof