Binomial-$\chi^2$ vector random fields

We introduce a new class of non-Gaussian vector random fields in space and/or time, which are termed binomial-$\chi^2$ vector random fields and include $\chi^2$ vector random fields as special cases. We define a binomial-$\chi^2$ vector random field as a binomial sum of squares of independent Gaussian vector random fields on a spatial, temporal, or spatio-temporal index domain. This is a second-order vector random field and has an interesting feature in that its finite-dimensional Laplace transforms are not determined by its own covariance matrix function, but rather by that of the underlying Gaussian one. We study the basic properties of binomial-$\chi^2$ vector random fields and derive some direct/cross covariances, which are based on the bivariate normal density, distribution, and related functions, for elliptically contoured and binomial-$\chi^2$ vector random fields.