On the admissibility of spherical spatial covariance functions in higher dimensions

Abstract

A mean zero spatial process X(t) on the n-dimensional Euclidean space Rn is isotropic if its covariance function (c.f.) is of the form: Rn(τ)=Cov[X(t),X(s)], where τ=|t−s|, t, s∈Rn, and Rn is an admissible function on R1. An isotropic spatial process X has a bounded range of dependence if sup{τ:Rn(τ)≠0}<∞. Here we consider a class of isotropic c.f.’s Rn(τ), n=1,2,… with bounded ranges of dependence, among which there are R1, the classical triangular c.f. on the real line (n=1), and R3, the spherical c.f. in dimension three (n=3). For each dimension n≥1, the admissibility of Rn(τ) as a c.f. in higher dimensions is studied. While it is well known that for each n≥1, Rn is a legitimate c.f. on Rm for all m≤n but it is shown that the considered Rn is not a legitimate c.f. on Rm when m>n. Thus the spherical c.f. R3 cannot be a c.f. on Rn when n>3. The issue of recognition of an isotropic c.f. on Rn is discussed, and simple procedures of constructing isotropic c.f.’s on Rn for every n≥1 are given. This article serves as one more reminder that caution must be taken concerning the legitimacy of a selected c.f. in the corresponding spatial dimensions.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1016/j.jspi.2021.07.008

Keywords

Characteristic function, Covariance function, Curse of dimensionality, Isotropic spatial process, Spatial process, Spherical covariance function

Publication Date

3-1-2022

Journal Title

Journal of Statistical Planning and Inference

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