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

1. Problem Statement and Context

The paper addresses the fixed-domain identifiability of covariance parameters for stationary Gaussian random fields, specifically within the Matérn class. In spatial statistics, data is often observed on a fixed bounded domain $D \subset \mathbb{R}^d$ with increasingly dense sampling (infill asymptotics).

The central question is: Under what conditions are two Matérn models with different parameters mutually singular (orthogonal) or equivalent?

• Microergodic Parameter: For a Matérn field with variance $\sigma^2$, range $\alpha$, and smoothness $\nu$, the combination $m = \sigma^2 \alpha^{2\nu}$ is known as the microergodic parameter.
• Known Results:
  • $d \leq 3$: Zhang [1] proved that if two models share the same microergodic parameter $m$ and smoothness $\nu$, they induce equivalent Gaussian measures. The parameters $\sigma^2$ and $\alpha$ cannot be consistently separated.
  • $d > 4$: Anderes [2] proved that if $m$ is the same but $\alpha$ differs, the measures are mutually singular. The parameters can be consistently separated.
• The Open Case: The critical dimension $d = 4$ remained unresolved. The paper aims to determine whether the accumulation of spectral information in 4D is sufficient to distinguish models with the same $m$ but different $\alpha$.

2. Methodology

The author resolves the $d=4$ case using a spectral approach that serves as an analogue to the physical-space increment methods used by Anderes.

A. Spectral Localization

Instead of working with physical space increments, the paper utilizes localized Fourier coefficients.

1. Localization: A smooth cutoff function $\chi \in C_c^\infty(D)$ is introduced to localize the field $Y(t)$ on the bounded domain.
2. Coefficients: Localized Fourier coefficients are defined as:
   $$X_k = \int_D \chi(t) e^{-ik \cdot t} Y(t) \, dt, \quad k \in \mathbb{N}^4.$$
3. Spectral Density: The variance of these coefficients, $v_j(k) = \mathbb{E}_j[|X_k|^2]$, acts as a smoothed version of the spectral density $f_j(\xi)$ near frequency $k$.

B. Asymptotic Analysis of Mismatch

The core of the proof relies on analyzing the relative variance mismatch $\delta_k$ between two models with parameters $(\sigma_1, \alpha_1)$ and $(\sigma_2, \alpha_2)$, where $\sigma_1^2 \alpha_1^{2\nu} = \sigma_2^2 \alpha_2^{2\nu} = m$ but $\alpha_1 \neq \alpha_2$.

• The spectral densities agree at the leading order ($|\xi|^{-2\nu-d}$).
• The first discrepancy appears at the next order. In dimension $d=4$, the mismatch behaves as:
  $$\delta_k = \frac{v_2(k)}{v_1(k)} - 1 \asymp |\alpha_1^2 - \alpha_2^2| |k|^{-2}.$$
• Critical Accumulation: The total information scale $L_N$ is the sum of squared mismatches over a frequency shell $\Lambda_N$:
  $$L_N = \sum_{k \in \Lambda_N} \delta_k^2 \asymp \sum_{k \in \Lambda_N} |k|^{-4}.$$
  In 4D, this sum diverges logarithmically ($L_N \asymp \log N$). This slow but unbounded accumulation is the key mechanism allowing identification.

C. Separation Statistic

A normalized quadratic statistic (Whittle score) is constructed:
$$T_N = \frac{1}{L_N} \sum_{k \in \Lambda_N} \delta_k \left( \frac{|X_k|^2}{v_1(k)} - 1 \right).$$

• Under Model 1, $\mathbb{E}[T_N] = 0$.
• Under Model 2, $\mathbb{E}[T_N] = 1$.
• Variance Control: The main technical challenge is that $X_k$ are not independent due to the boundary effects of the bounded domain. The author proves that the off-diagonal covariances decay rapidly (due to the smoothness of $\chi$), ensuring that $\text{Var}(T_N) \asymp L_N^{-1} \asymp (\log N)^{-1}$.
• By selecting a sparse subsequence $N_s$ such that $\sum L_{N_s}^{-1} < \infty$, the Borel-Cantelli lemma ensures almost sure convergence: $T_{N_s} \to 0$ under $P_1$ and $T_{N_s} \to 1$ under $P_2$.

3. Key Contributions

1. Resolution of the Critical Case: The paper proves that for $d=4$, two stationary Matérn fields with the same microergodic parameter $m$ but different range parameters $\alpha$ induce mutually singular Gaussian measures on any countable dense observation set.
2. Spectral Analogue of Increment Methods: It establishes a rigorous connection between physical-space higher-order increments (Anderes' method) and spectral-space localized probing. It shows that the "second irregular coefficient" in physical space corresponds to the "first non-vanishing spectral mismatch" in the frequency domain.
3. Handling Off-Diagonal Dependencies: Unlike previous work on bounded domains (e.g., Bolin & Kirchner) which relied on exact diagonalization via Dirichlet eigenfunctions, this work handles the non-stationary nature of the localized coefficients by proving rapid decay of off-diagonal covariances.
4. Unified Framework: The approach is generalizable to $d \geq 4$, recovering the orthogonality results for $d > 4$ while explaining the logarithmic divergence specific to $d=4$.

4. Main Results

• Theorem 1: Let $D \subset \mathbb{R}^4$ be a bounded domain and $S \subset D$ be a countable dense set. If two Matérn models have the same smoothness $\nu$ and microergodic parameter $m$, but different range parameters $\alpha_1 \neq \alpha_2$, then the laws of the processes restricted to $S$ are mutually singular ($P_1^S \perp P_2^S$).
• Path Space Singularity: The singularity holds not just on discrete grids but on the space of continuous functions $C(D)$.
• Simulation Evidence: Numerical simulations confirm that the statistic $T_N$ successfully separates models with different $\alpha$ values in $d=4$, with the empirical mean converging to 0 or 1 as predicted.
• Estimation: The paper demonstrates that a Whittle pseudo-likelihood estimator based on these spectral coefficients can consistently recover the range parameter $\alpha$ in $d=4$, validating the theoretical identifiability.

5. Significance

• Theoretical Completeness: This work closes a long-standing gap in the theory of Gaussian random fields, completing the dimension-dependent phase transition picture for Matérn fields ($d \leq 3$: equivalence; $d \geq 4$: orthogonality).
• Statistical Implications: It confirms that in 4-dimensional spatial data (relevant in certain physical and environmental applications), the range parameter $\alpha$ is consistently estimable from a single realization, provided the sampling is dense enough.
• Methodological Innovation: The "localized spectral probing" framework offers a new tool for studying identifiability in non-stationary or bounded-domain settings where traditional spectral methods (assuming infinite domains) fail. It bridges the gap between operator-based approaches (fractional Laplacians) and classical covariance analysis.
• Practical Application: The results support the use of Whittle likelihood estimation for Matérn fields in 4D, suggesting that standard spectral estimation techniques remain valid even in the critical dimension where information accumulation is only logarithmic.