微可积性意味着在$\mathbb{R}^4$有界域上Matérn场的正交性
Microergodicity implies orthogonality of Matérn fields on bounded domains in $\mathbb{R}^4$
AI总结：
本文研究Matérn场在$\mathbb{R}^4$有界域上的正交性，证明当维度为4时，微可积参数相同但其他参数不同的Matérn模型对应的高斯测度是互奇异的。
摘要：
Matérn随机场是空间统计中最广泛使用的模型之一。固定域下，平稳Matérn高斯随机场的协方差参数可识别性表现出维度依赖的相变。对于已知的光滑度$ν$，Zhang（2004）证明当$d\le3$时，具有相同微可积参数$m=σ^2α^{2ν}$的两个Matérn模型在有界域上诱导等价的高斯测度，而Anderes（2010）证明当$d>4$时，参数不同的相应测度是互奇异的。对于平稳Matérn模型的临界情况$d=4$一直未解决。本文解决了这一情况。设$d=4$，考虑在$\mathbb R^4$上参数为$(σ_1,α_1)$和$(σ_2,α_2)$的两个平稳Matérn模型，满足$σ_1^2α_1^{2ν}=σ_2^2α_2^{2ν}$，且$α_1\neq α_2$。证明在任何有界观测域上，对应的高斯测度在每一个可数稠密观测集上以及关联的连续函数路径空间上都是互奇异的。本文的方法可以视为Anderes（2010）高阶增量方法的谱类比。尽管Anderes通过物理空间中的归一化二次变分分离第二不规则协方差系数，我们则通过局部化傅里叶系数检测第一个非零高频谱不匹配，并使用归一化Whittle分数来识别参数。更广泛地说，本文用于检测高斯随机场中细微协方差差异的局部化谱探测框架可能对研究其他空间模型的可识别性和估计具有帮助。
英文摘要：
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.
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