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

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.

• The paper establishes that in R⁴, microergodically matched Matérn fields with differing range parameters are mutually singular.
• Using spectral localization methods, the authors construct a quadratic statistic to robustly distinguish between models via logarithmic divergence.
• Simulations confirm that the diagonal Whittle pseudo-likelihood reliably recovers the range parameter, supporting high-dimensional spatial inference.

Microergodicity and Orthogonality of Matérn Fields in $\mathbb{R}^4$

Matérn Fields and the Microergodic Parameter

The paper examines stationary Matérn Gaussian random fields, a class widely employed in spatial statistics owing to their flexible covariance structure governed by the variance $\sigma^2$, range $\alpha$, and smoothness $\nu$. The covariance function on $\mathbb{R}^d$ is given by

$$K(h) = \sigma^2 \frac{(\alpha |h|)^\nu}{2^{\nu-1} \Gamma(\nu)} \mathcal{K}_\nu(\alpha |h|),$$

with spectral density

$$f_{\sigma,\alpha,\nu}(\xi) \propto \sigma^2 \alpha^{2\nu} (\alpha^2 + |\xi|^2)^{-(\nu + d/2)}.$$

The microergodic parameter $m = \sigma^2 \alpha^{2\nu}$ governs the leading-order spectral tail and dominates fixed-domain (infill) asymptotic inference. In dimensions $d \leq 3$, equivalence of Gaussian measures induced by two Matérn fields with different $(\sigma, \alpha)$ but identical $m$ is established, precluding consistent separation of $\sigma^2$ and $\alpha$ from dense data [Zhang 2004]. For $d > 4$, mutual singularity holds whenever the parameters differ [Anderes 2010], allowing full identifiability.

Resolution of the Critical Case: $d=4$

The critical and previously unresolved case is $d = 4$. The paper's main result rigorously establishes mutual singularity of Matérn measures with equal smoothness $\nu$ and microergodic parameter $m$, but distinct range parameters $\alpha_1 \ne \alpha_2$, observed on any bounded domain and countable dense set. That is, for stationary Matérn models on $\mathbb{R}^4$,

$$\sigma_1^2 \alpha_1^{2\nu} = \sigma_2^2 \alpha_2^{2\nu}, \quad \alpha_1 \ne \alpha_2,$$

the induced probability measures are mutually singular. Thus, in $d=4$, $\sigma^2$ and $\alpha$ are asymptotically distinguishable from dense fixed-domain data.

Spectral Probing: Technical Approach

The proof employs a spectral localization method, constructing localized Fourier coefficients $X_k$ from the field restricted to a bounded domain. Under microergodic matching, the spectral densities coincide to leading order, but a subtle mismatch appears in the next-order term,

$$f_2(\xi) - f_1(\xi) \sim |\xi|^{-2p-2}, \quad p = \nu + 2,$$

and their ratio decays as $|\xi|^{-2}$. In $d=4$, the sum $\sum_{|k|\leq N} |k|^{-4} \asymp \log N$ accumulates only logarithmically. This marginal divergence provides a detectable statistical signal.

A quadratic statistic is constructed:

$$T_N = \frac{1}{L_N} \sum_{k \in \Lambda_N} \delta_k \left( \frac{|X_k|^2}{v_1(k)} - 1 \right), \quad \delta_k = \frac{v_2(k)}{v_1(k)} - 1,$$

with normalization $L_N = \sum_{k\in\Lambda_N} \delta_k^2$. As $N \to \infty$, $L_N \asymp \log N$. Under models 1 and 2, $E_1[T_N] = 0$ and $E_2[T_N] = 1$, providing asymptotic separation. Despite dependence among $\{X_k\}$, rapid decay of localization-induced off-diagonal terms allows variance control, yielding almost sure convergence.

Figure 1: Empirical distribution of $T_N$ showing concentration near $0$ under model 1 and near $1$ under model 2 for $\alpha_1 = 1$, $\alpha_2 = 2$.

Figure 2: Empirical distribution of $T_N$ for $\alpha_1 = 1$, $\alpha_2 = 1.2$; separation persists but variance increases due to weaker spectral separation.

Score Statistic Interpretation and Likelihood Formulation

The statistic $T_N$ closely corresponds to the Gaussian score in the $\alpha^2$ direction for the localized spectral law. Under diagonal approximation,

$$T_N = \frac{S_N}{L_N}, \quad S_N = \sum_{k \in \Lambda_N} \delta_k \left( \frac{|X_k|^2}{v_1(k)} - 1 \right),$$

the log-likelihood ratio expands as $S_N - \frac{1}{2} L_N + o(L_N)$. Chebyshev's inequality and variance decay ($\operatorname{Var}(T_N) \lesssim 1 / L_N$) guarantee separation along sparse sequences.

Whittle-Type Estimation

A simulation-based procedure demonstrates parameter estimation via diagonal Whittle pseudo-likelihood from grid Fourier coefficients. For fixed smoothness, the estimator accurately recovers the range parameter even in $d=4$, confirming that localized frequency information retains parameter identifiability in the critical dimension.

Figure 3: Monte Carlo distribution of the Whittle estimator $\widehat{\alpha}$ with the true value $\alpha = 3$ indicated by the dashed line.

Comparison to Previous Work and Theoretical Implications

The methodology connects with the increment-based approach of Anderes, which isolates second-order irregular covariance coefficients via high-order increments in physical space. The spectral analogue developed here detects the first nonvanishing high-frequency mismatch. The criticality of $d=4$ arises because the information scale diverges only logarithmically.

The result aligns with recent operator-based analyses for Matérn-type fields on bounded domains [BoliKirc2023], where the Feldman–Hajek square-summability criterion yields equivalence if and only if range parameters coincide for $d \geq 4$.

Practical Implications and Future Directions

The findings demonstrate that in four-dimensional spatial settings, dense observations allow separation of variance and range parameters, fundamentally impacting inference in high-dimensional geostatistics. The spectral probing framework opens avenues for high-frequency aggregation-based parameter estimation and model selection for Gaussian random fields and related spatial models.

Potential future directions include:

• Analysis and consistency guarantees for Whittle likelihood estimators in $d=4$ domains.
• Extension of spectral localization for equivalence/orthogonality classification in non-Euclidean or manifold domains.
• Generalization to other covariance families exhibiting critical dimension phenomena.

Conclusion

The paper resolves the critical $d=4$ case for Matérn random fields, establishing mutual singularity of microergodically matched models with differing range. The spectral probing methodology highlights logarithmic accumulation as the decisive mechanism and enables both theoretical separation and practical estimation. These results serve as a technical foundation for spatial inference in high-dimensional settings, informing future statistical methodology for Gaussian fields.