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

Abstract

Mat\'ern random fields are one of the most widely used classes of models in spatial statistics. The fixed-domain identifiability of covariance parameters for stationary Mat\'ern Gaussian random fields exhibits a dimension-dependent phase transition. For known smoothness $\nu$, Zhang \cite{Zhang2004} showed that when $d\le3$, two Mat\'ern models with the same microergodic parameter $m=\sigma^2\alpha^{2\nu}$ 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 Mat\'ern models has remained open. We resolve this case. Let $d=4$ and consider two stationary Mat\'ern models on $\mathbb R^4$ with parameters $(\sigma_1,\alpha_1)$ and $(\sigma_2,\alpha_2)$ satisfying \[ \sigma_1^2\alpha_1^{2\nu}=\sigma_2^2\alpha_2^{2\nu}, \qquad \alpha_1\neq \alpha_2. \] 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.