K-combined random fields: Basic properties and stochastic orderings

This paper introduces the K-combined vector random field, whose finite-dimensional characteristic functions are made up of certain power functions and whose finite-dimensional density functions are comprised of the modified Bessel functions of the second type. It is an elliptically contoured vector random field, contains K-differenced vector random field of Alsultan and Ma (2019) as a special case, and possesses all orders of moments. It is fully characterized by its mean vector function and its covariance matrix function, just like a Gaussian vector random field. With various selection of its parameters, its finite-dimensional distributions may have heavy tails or thin tails, comparing with a Gaussian one, and thus it provides a potential model for applications. We also investigate the usual stochastic ordering, the convex ordering, and the peakedness ordering of K-combined random fields and of multivariate K-combined distributions, with necessary and/or sufficient conditions derived.