A matrix-valued Schoenberg's problem and its applications

In this paper we present a criterion for positive definiteness of the matrix-valued function f (t) := exp(-|t|alpha[B+ + B- sign(t)]), where alpha is an element of (0, 2] and B1 are real symmetric and antisymmetric d x d matrices. We also find a criterion for positive definiteness of its multidimensional generalization f (t) := exp(- fSd-1 |tTs|alpha[B+ + B- sign(tTs)] d Lambda(s)) where Lambda is a finite measure on the unit sphere Sd-1 subset of Rd under a more restrictive assumption that B1 commute and are normal. The associated stationary Gaussian random field may be viewed as as a generalization of the univariate fractional OrnsteinUhlenbeck process. This generalization turns out to be particularly useful for the asymptotic analysis of Rd-valued Gaussian random fields. Another possible application of these findings may concern variogram modelling and general stationary time series analysis.