Tauberian and Abelian theorems for correlation function of a homogeneous isotropic random field

We prove theorems of Tauberian and Abelian types for nonintegrable correlation functions of homogeneous isotropic random fields and use them to study asymptotic distributions of local functionals of Gaussian fields.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1652–1664, December, 1991.