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

TL;DR

This paper investigates the conditions under which a covariance function can be realized by a real-valued random field with values in specific subsets of the real line, extending classical results to more constrained cases.

Contribution

It provides necessary and sufficient conditions for covariance functions to correspond to random fields valued in various subsets of the real line, including intervals and finite sets.

Findings

Conditions reduce to symmetry and positive semidefiniteness for real or integer-valued fields.

More restrictive conditions are needed for fields valued in intervals or two-point sets.

Characterizations are extended to semivariograms, higher-order moments, and multivariate fields.

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 $\rho: \X \times \X \to \R$ 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 $\rho$ 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 $\rho$ are no longer enough to guarantee the existence of a random field valued in ${\cE}$ and having $\rho$ as its non-centered covariance. Similar characterizations are obtained for semivariograms and higher-order spatial moments, as well as for multivariate random fields.