Limit theorems for anisotropic functionals of stationary Gaussian fields with Gneiting covariance function

We study non-linear additive functionals of stationary Gaussian fields over anisotropically growing domains in $\mathbb{R}^d$, including spatiotemporal settings, and establish Gaussian and non-Gaussian limit theorems under non-separable covariance structures. We characterize the regimes in which the normalized functionals converge either to a Gaussian distribution or to a $2$-domain Rosenblatt distribution, depending on precise long-range dependence conditions. Our analysis covers covariance functions from the Gneiting class, which provides a canonical family of non-separable spatiotemporal models. A key structural result shows that such covariances are asymptotically separable in a precise cumulant sense, allowing us to identify explicitly the limiting distributions without imposing additional spectral assumptions. These results extend existing spatiotemporal limit theorems beyond separable and short-memory frameworks and provide a unified description of anisotropic long-range dependence phenomena.

💡 Research Summary

The paper investigates the asymptotic behavior of nonlinear additive functionals of stationary Gaussian random fields when the observation domain expands anisotropically in ℝⁿ, with a particular focus on the Gneiting class of non‑separable space‑time covariance functions. Let B(x₁,x₂) be a centered, unit‑variance Gaussian field indexed by (x₁∈ℝ^{d₁}, x₂∈ℝ^{d₂}) and let φ be a non‑constant square‑integrable function admitting a Hermite expansion φ(u)=∑_{q≥R}a_q H_q(u). The Hermite rank R (the smallest q with a_q≠0) is assumed to be at least 2. For convex, compact sets D₁⊂ℝ^{d₁} and D₂⊂ℝ^{d₂} the functional of interest is

 Y(t)=∫_{t₁D₁×t₂D₂} φ(B(x₁,x₂)) dx₁dx₂, t=(t₁,t₂)→∞,

with the standardization rY(t)=(Y(t)−E Y(t))/√Var Y(t).

The covariance is taken from the Gneiting family:

 C(x₁,x₂)=C₂(‖x₂‖)·C₁(‖x₁‖/C₂(‖x₂‖)),

where C₁ and C₂ are radial, C₁ is completely monotone, C₂>0 and has a completely monotone derivative. Both functions are assumed to be regularly varying at infinity with indices ρ₁>0 and ρ₂>0 respectively (i.e., C_i(r)≈r^{-ρ_i}L_i(r) with slowly varying L_i). The anisotropic growth rates satisfy the technical condition lim_{t→∞} t₁ c₂(t₂)=∞, ensuring that the effective spatial rescaling induced by the Gneiting structure diverges sufficiently fast.

Three main theorems are proved.

Theorem 1.1 (Gaussian regime, short‑range spatial dependence). If C₁∈L²(ℝ^{d₁}) (short‑range dependence) while C₂ may be long‑range but satisfies the growth condition, then rY(t) converges in distribution to a standard normal N(0,1). Moreover Var Y(t)∼ℓ t₁^{d₁}t₂^{d₂}, where ℓ is an explicit constant involving the volumes of D₁, D₂ and the integrals of C₁ and C₂.

Theorem 1.2 (Gaussian regime, long‑range spatial dependence). When C₁ exhibits long‑range dependence with exponent ρ₁ such that 2ρ₁<d₁ and C₂ is short‑range, the variance grows like ℓ₂ t₂^{d₂} t₁^{2d₁−Rρ₁}. Despite the super‑linear growth, the normalized functional still satisfies a central limit theorem, i.e., rY(t)→N(0,1).

Theorem 1.3 (Non‑Gaussian Rosenblatt regime). For Hermite rank R=2 and both C₁ and C₂ long‑range with exponents ρ₁, ρ₂ satisfying 2ρ₁<d₁ and ρ₂<d₁d₂/(2(d₁−ρ₁)), the limit is non‑Gaussian. Specifically, rY(t) converges to a 2‑domain Rosenblatt random variable H_{ρ₁,ρ₂}, which lives in the second Wiener chaos and has cumulants that match those of a double integral with kernel determined by the asymptotic behavior of C₁ and C₂. The variance in this case behaves as ℓ₃ t₁^{2d₁−Rρ₁} t₂^{2d₂−Rρ₂}.

The proofs rely on Malliavin calculus combined with the Fourth Moment Theorem. The k‑th cumulant κ_k(t) of rY(t) is expressed as a multiple integral involving the covariance kernel; it can be written as κ_k(t)=2^{k-1}(k-1)! c_k(t). By exploiting regular variation, the authors show that c_k(t)→0 for all k≥3 in the Gaussian regimes, while in the Rosenblatt regime c_k(t) converges to a non‑zero limit for k=2,4,…, reproducing the cumulant structure of the Rosenblatt distribution. A crucial structural lemma proves that Gneiting covariances become asymptotically separable in the sense that C(x₁,x₂)≈C₁(x₁)C₂(x₂) for large arguments, which allows the authors to transfer existing limit theorems for separable covariances to the non‑separable setting without additional spectral assumptions.

The paper situates its contributions relative to prior work: earlier results required either separable covariances, restrictive spectral conditions, or isotropic growth of the observation window. By handling fully non‑separable Gneiting covariances and arbitrary anisotropic scaling, the authors provide a unified framework that captures both Gaussian and non‑Gaussian limits. The results have immediate implications for statistical inference on spatiotemporal data modeled with Gneiting covariances, especially when long‑range dependence and anisotropic sampling are present. Potential extensions include critical cases (logarithmic corrections) and higher Hermite ranks (R>2).