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|α[B++B−sign(t)]), where α∈(0,2] and B± are real symmetric and antisymmetric d×d matrices. We also find a criterion for positive definiteness of its multidimensional generalization f(t):=exp(−∫Sd−1|t⊤s|α[B++B−sign(t⊤s)]dΛ(s)) where Λ is a finite measure on the unit sphere Sd−1⊂Rd under a more restrictive assumption that B± commute and are normal. The associated stationary Gaussian random field may be viewed as as a generalization of the univariate fractional Ornstein-Uhlenbeck 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.