On the compatibility between the spatial moments and the codomain of a real random field

While any symmetric and positive semidefinite mapping can be the non-centered covariance of a Gaussian random field, it is known that these conditions are no longer sufficient when the random field is valued in a two-point set. The question therefore arises of what are the necessary and sufficient conditions for a mapping $ρ: \X \times \X \to \R$ to be the non-centered covariance of a random field with values in a subset ${\cE}$ of $\R$. Such conditions are presented in the general case when ${\cE}$ is a closed subset of the real line, then examined for some specific cases. In particular, if ${\cE}=\R$ or $\Z$, it is shown that the conditions reduce to $ρ$ being symmetric and positive semidefinite. If ${\cE}$ is a closed interval or a two-point set, the necessary and sufficient conditions are more restrictive: the symmetry, positive semidefiniteness, upper and lower boundedness of $ρ$ are no longer enough to guarantee the existence of a random field valued in ${\cE}$ and having $ρ$ as its non-centered covariance. Similar characterizations are obtained for semivariograms and higher-order spatial moments, as well as for multivariate random fields.

💡 Research Summary

The paper addresses a fundamental problem in spatial statistics: determining when a given function ρ on a set of locations X can serve as the non‑centered covariance of a real‑valued random field whose values are restricted to a subset E of the real line. While Schoenberg’s classical result guarantees that any symmetric positive‑semidefinite (PSD) kernel is a covariance for a Gaussian field when the codomain is the whole real line, this condition is no longer sufficient when the field is forced to take values in a proper subset of ℝ (for example, a binary field {‑1, 1}). The authors introduce a novel analytical tool – the “gap” – to capture the additional constraints imposed by the codomain.

Two types of gaps are defined. The γ‑gap of a real matrix Λ with respect to a set E is γ(Λ,E)=inf_{z∈Eⁿ} zΛzᵀ, i.e., the smallest quadratic form attainable by vectors whose components lie in E. The η‑gap is η(Λ,E)=sup_{z∈Eⁿ}½∑{k,ℓ}λ{kℓ}(z_k−z_ℓ)², i.e., the largest possible sum of weighted squared differences. Both gaps enjoy linearity, convexity, scaling, and monotonicity properties, and they extend naturally to functions via appropriate integral formulations.

The main theoretical contributions are three theorems.
Theorem 1 (closed codomain) states that a mapping ρ:X×X→ℝ is the non‑centered covariance of an E‑valued random field if and only if (i) ρ is symmetric and (ii) for every finite collection of points {x₁,…,x_n}⊂X and every real matrix Λ, the inequality ⟨Λ,R⟩≥γ(Λ,E) holds, where R_{kℓ}=ρ(x_k,x_ℓ) and ⟨·,·⟩ denotes the trace inner product. In other words, the usual PSD condition is strengthened by a lower bound given by the γ‑gap.
Theorem 2 (bounded but not closed codomain) shows that when E is bounded yet not closed, a function ρ can be approximated pointwise by covariances of E‑valued fields precisely when the same symmetry and gap inequality are satisfied. This result provides a constructive approximation framework for practical situations where the codomain is an open interval or a finite set lacking its boundary points.
Theorem 3 (semivariograms) deals with zero‑drift fields. It asserts that a symmetric function g:X×X→ℝ is a valid semivariogram of an E‑valued field if and only if for every symmetric matrix Λ and every point set {x_i}, the inequality ⟨Λ,G⟩≤η(Λ,E) holds, where G_{kℓ}=g(x_k,x_ℓ). The η‑gap thus supplies an upper bound on admissible semivariograms, complementing the lower bound supplied by the γ‑gap for covariances.

The authors examine several concrete codomains. When E=ℝ or E=ℤ, the γ‑gap reduces to zero, so the conditions collapse to the classical symmetry‑and‑PSD requirement. For a closed interval