Tauberian and Abelian theorems for long-range dependent random fields

TL;DR

This paper reviews Abelian and Tauberian theorems relevant to long-range dependent random fields, highlighting their applications and limitations in understanding asymptotic behaviors of covariance functions, spectral densities, and averaged functionals.

Contribution

It provides a comprehensive framework for asymptotic analysis of long-range dependent random fields and demonstrates their application to new covariance function examples.

Findings

Framework for asymptotic behavior of covariance functions

Application to new examples of covariance functions

Limitations of theorems in certain cases

Abstract

This paper surveys Abelian and Tauberian theorems for long-range dependent random fields. We describe a framework for asymptotic behaviour of covariance functions or variances of averaged functionals of random fields at infinity and spectral densities at zero. The use of the theorems and their limitations are demonstrated through applications to some new and less-known examples of covariance functions of long-range dependent random fields.