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.

翻译：Matérn随机场是空间统计学中应用最广泛的模型类别之一。平稳Matérn高斯随机场的协方差参数在固定域上的可识别性呈现维度依赖的相变现象。对于已知光滑度参数$ν$，Zhang \cite{Zhang2004} 证明当$d\le3$时，具有相同微遍历参数$m=σ^2α^{2ν}$的两个Matérn模型在有界域上诱导等价的高斯测度；而Anderes \cite{Anderes2010} 则证明当$d>4$时，参数不同的对应测度相互奇异。平稳Matérn模型的临界情形$d=4$尚未解决。本文解决了该问题。设$d=4$，考虑$\mathbb R^4$上两个参数为$(σ_1,α_1)$和$(σ_2,α_2)$的平稳Matérn模型，满足\[ σ_1^2α_1^{2ν}=σ_2^2α_2^{2ν}, \qquad α_1\neq α_2. \]我们证明，在任意有界观测域上，对应的高斯测度在每个可数稠密观测点集以及关联的连续函数路径空间上均相互奇异。该方法可视为Anderes \cite{Anderes2010}高阶增量法的谱域类比。Anderes通过物理空间中的重整化二次变分分离第二不规则协方差系数，而本文通过局域傅里叶系数检测首个非零高频谱失配，并利用归一化Whittle得分识别参数。更广泛地，本文用于检测高斯随机场中细微协方差差异的局域谱探测框架，可能对其他空间模型的可识别性与估计研究具有参考价值。