Abelian and Tauberian theorems for random fields on two-point homogeneous spaces

Abstract

Abstract. We consider centered mean-square continuous random fields for which the variance of increments between two points depends only on the distance between these points. Relations between the asymptotic behavior of the variance of increments near zero and the asymptotic behavior of the spectral measure of the field near infinity are investigated. We prove several Abelian and Tauberian theorems in terms of slowly varying functions.

Series expansions among weighted Lebesgue function spaces and applications to positive definite functions on compact two-point homogeneous spaces

“…The book by [30] contains a comprehensive introduction. Random fields on two-point homogeneous spaces have been inspected by [28] and [29]. Generalizations have been considered by [19].…”

“…It is enough to observe that the Jacobi polynomials are not orthonormal. Then, using Remark 4.1 and the Equations ( 15), ( 22)-( 24), we obtain (29) where…”