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

Xavier Emery, Christian Lantuéjoul

TL;DR

The paper addresses the problem of when a function $\rho$ can serve as the non-centered covariance of a real-valued random field whose values lie in a closed (or compact) subset $\mathcal{E}$ of $\mathbb{R}$. It develops a unified gap-inequality framework built around the $\gamma$-gap and the $\eta$-gap to provide necessary and sufficient conditions in both discrete (matrix) and continuous (kernel) settings, recovering classical results for ${\mathcal{E}}=\mathbb{R}$ or $\mathbb{Z}$ and revealing stricter constraints for bounded intervals or two-point codomains. The work extends to semivariograms, higher-order spatial moments, and multivariate random fields, and connects realizability to convex-analytic objects like the convex hull of rank-one kernels and completely positive/copositive cones. It also leverages extension theorems (e.g., Daniell–Kolmogorov) to guarantee the existence of admissible random fields under the proposed conditions, with implications for spatial statistics and stochastic modeling under codomain constraints.

Abstract

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.

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

TL;DR

The paper addresses the problem of when a function can serve as the non-centered covariance of a real-valued random field whose values lie in a closed (or compact) subset of . It develops a unified gap-inequality framework built around the -gap and the -gap to provide necessary and sufficient conditions in both discrete (matrix) and continuous (kernel) settings, recovering classical results for or and revealing stricter constraints for bounded intervals or two-point codomains. The work extends to semivariograms, higher-order spatial moments, and multivariate random fields, and connects realizability to convex-analytic objects like the convex hull of rank-one kernels and completely positive/copositive cones. It also leverages extension theorems (e.g., Daniell–Kolmogorov) to guarantee the existence of admissible random fields under the proposed conditions, with implications for spatial statistics and stochastic modeling under codomain constraints.

Abstract

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 to be the non-centered covariance of a random field with values in a subset of . Such conditions are presented in the general case when is a closed subset of the real line, then examined for some specific cases. In particular, if or , it is shown that the conditions reduce to being symmetric and positive semidefinite. If 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 are no longer enough to guarantee the existence of a random field valued in and having as its non-centered covariance. Similar characterizations are obtained for semivariograms and higher-order spatial moments, as well as for multivariate random fields.
Paper Structure (18 sections, 34 theorems, 70 equations)

This paper contains 18 sections, 34 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Background material
  3. Definitions
  4. Properties
  5. Gap inequalities in a discrete setting
  6. Characterization of non-centered covariance functions
  7. Characterization of semivariograms
  8. Characterization of spatial moments beyond covariance functions
  9. Multivariate random fields
  10. Gap inequalities in a continuous setting
  11. Concluding remarks
  12. A look at particular codomains
  13. Random fields valued in $\mathbb{R}$ or $\mathbb{Z}$
  14. Random fields valued in $\mathbb{Z} \smallsetminus \{0\}$
  15. Binary random fields
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\boldsymbol{\Lambda}$ be a real symmetric positive semidefinite matrix. Then $\gamma(\boldsymbol{\Lambda},{\@fontswitch\mathcal{E}}) \geq 0$, and $\gamma(\boldsymbol{\Lambda},{\@fontswitch\mathcal{E}}) = 0$ as soon as $0 \in {\@fontswitch\mathcal{E}}$.

Theorems & Definitions (80)

  • Definition 1: Trace inner product
  • Definition 2: $\gamma$-gap of a real square matrix
  • Definition 3: $\gamma$-gap of a multidimensional array
  • Definition 4: $\eta$-gap of a real square matrix
  • Definition 5: Hilbert space of square integrable functions of $\mathbb{X}^2$
  • Definition 6: $\gamma$-gap of a real function
  • Definition 7: $\eta$-gap of a real function
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • ...and 70 more