Covariance Matrix Functions of Vector χ^2 Random Fields in Space and Time

This paper introduces vector (multivariate, or multiple) χ² and Rayleigh random functions or random fields on a spatial, temporal, or spatio-temporal index domain, and explores their basic properties. Formulated as a sum of squares of independent Gaussian random functions, a vector χ² random function has an interesting feature that its finite-dimensional Laplace transforms are not determined by its own covariance matrix, but by that of the underlying Gaussian one. With the conditionally negative definite matrix as an important building block, this paper constructs a class of vector χ² random functions, from whose covariance matrices one can easily identify those of the underlying Gaussian one so that the resulting vector χ² random function can be easily simulated and analyzed.