Tauberian and Abelian Theorems for Long-range Dependent Random Fields

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.In this paper we study asymptotic properties of spectral and covariance functions of random fields. We investigate the connection between (a) the behaviour of covariance functions or variances of averaged functionals of random fields at infinity and (b) the behaviour of spectral distribution functions at zero. In the terminology of the integral transforms the statement "(b) implies (a)" is a theorem of Abelian type and the statement "(a) implies (b)" is a Tauberian theorem.Abelian and Tauberian theorems are not only of their own interest but also have numerous applications in asymptotic problems of probability theory and statistics

On a class of minimum contrast estimators for Gegenbauer random fields

“…A detailed discussion on relations between local specifications of spectral functions and the tail behaviour of auto-covariance functions of long range dependent random fields can be found in Leonenko and Olenko (2013). …”

On LSE in regression model for long-range dependent random fields on spheres

“…Remark 4. By Tauberian and Abelian theorems, see [28], for L 0 (·) and L(·) given in Assumptions 1 and 2 it holds L 0 (r) ∼ L(r), r → +∞.…”

Estimation of cyclic long‐memory parameters

“…It is known that the phenomenon of long‐range dependence is related to singularities of spectral densities; see Leonenko and Olenko (). The majority of publications study the case when spectral densities are unbounded at the origin.…”

“…KEYWORDS estimators of parameters, filter, Gegenbauer-type spectral densities, seasonal/cyclic long memory, stochastic process, wavelet transformation ∞ 0 |B(r)|dr = +∞, or, more precisely, as a hyperbolic asymptotic behavior of B(·). It is known that the phenomenon of long-range dependence is related to singularities of spectral densities; see Leonenko and Olenko (2013). The majority of publications study the case when © 2019 Board of the Foundation of the Scandinavian Journal of Statistics 104 wileyonlinelibrary.com/journal/sjos Scand J Statist.…”