Microergodicity implies orthogonality of Matérn fields on bounded domains in $\mathbb{R}^4$

2026-03-25

AI Summary

This study resolves a long-standing open problem concerning the equivalence or singularity of Gaussian measures induced by Matérn Gaussian random fields on a four-dimensional bounded domain, under fixed microergodic parameters. Specifically, for fields sharing the same smoothness parameter but differing in range parameters, the work establishes—for the first time—the mutual singularity of their induced Gaussian measures. To achieve this, the authors introduce a localized spectral probing framework that identifies covariance discrepancies through high-frequency spectral mismatch, departing from classical physical-space quadratic variation methods. Instead, the approach leverages local Fourier coefficient analysis, normalized Whittle scores, and higher-order spectral increment techniques. Notably, when σ₁²α₁²ᵛ = σ₂²α₂²ᵛ with α₁ ≠ α₂, the corresponding measures are shown to be singular both on the space of continuous sample paths and under dense observation schemes, thereby establishing a sharp criterion for measure singularity in the critical case d = 4.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Web Mining and Content Analysis: Web measurements
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Matérn random fields are one of the most widely used classes of models in spatial statistics. The fixed-domain identifiability of covariance parameters for stationary Matérn Gaussian random fields exhibits a dimension-dependent phase transition. For known smoothness $ν$, Zhang \cite{Zhang2004} showed that when $d\le3$, two Matérn models with the same microergodic parameter $m=σ^2α^{2ν}$ induce equivalent Gaussian measures on bounded domains, while Anderes \cite{Anderes2010} proved that when $d>4$, the corresponding measures are mutually singular whenever the parameters differ. The critical case $d=4$ for stationary Matérn models has remained open. We resolve this case. Let $d=4$ and consider two stationary Matérn models on $\mathbb R^4$ with parameters $(σ_1,α_1)$ and $(σ_2,α_2)$ satisfying \[ σ_1^2α_1^{2ν}=σ_2^2α_2^{2ν}, \qquad α_1\neq α_2. \] We prove that the corresponding Gaussian measures on any bounded observation domain are mutually singular on every countable dense observation set, and on the associated path space of continuous functions. Our approach can be viewed as a spectral analogue of the higher-order increment method of Anderes \cite{Anderes2010}. Whereas Anderes isolates the second irregular covariance coefficient through renormalized quadratic variations in physical space, we detect the first nonvanishing high-frequency spectral mismatch via localized Fourier coefficients and use a normalized Whittle score to identify parameters. More broadly, the localized spectral probing framework used here for detecting subtle covariance differences in Gaussian random fields may be useful for studying identifiability and estimation in other spatial models.

Problem

• Microergodicity
• Matérn fields
• Gaussian measures
• Identifiability
• Spatial statistics

Innovation

• microergodicity
• Matérn random fields
• mutual singularity
• localized spectral analysis
• Whittle likelihood