On the admissibility of spherical spatial covariance functions in higher dimensions


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.



Document Type





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

Publication Date


Journal Title

Journal of Statistical Planning and Inference