Vector random fields on the arccos-quasi-quadratic metric space

Abstract

An arccos-quasi-quadratic metric is defined as the composition of arccosine and quasi-quadratic functions over a subset of R d + 1 such as a ball, an ellipsoid, a simplex or a corner of the d-cube, a bounded solid cone, a bounded solid hyperboloid, or their surfaces. This metric is not only conditionally negative definite (or of negative type) but also a measure definite kernel, and the metric space incorporates several important cases in a unified framework so that we are able to study metric-dependent random fields on different metric spaces in a unified manner. This paper constructs the Gaussian vector random field on the arccos-quasi-quadratic metric space via an infinite series expression based on spherical harmonics, introduces a class of elliptically contoured vector random fields with metric-dependent covariance matrix functions that possess the ultraspherical polynomial expressions, and studies their sample path properties. In particular, the property of strong local nondeterminism and Hölder continuity are established, and Abelian and Tauberian theorems are derived.