Asymptotic behaviour of functionals of cyclical long-range dependent random fields

TL;DR

This paper studies the asymptotic properties of functionals of cyclical long-range dependent random fields, revealing new covariance behaviors and establishing limit theorems for weighted functionals with applications to understanding their long-term behavior.

Contribution

It introduces new examples of covariance functions with non-regular varying asymptotics and derives limit theorems for weighted functionals of cyclical long-range dependent fields.

Findings

Covariance functions can have non-regular varying asymptotics.

Variances of averaged functionals are regularly varying.

Limit theorems for weighted functionals are established.

Abstract

Long-range dependent random fields with spectral densities which are unbounded at some frequencies are investigated. We demonstrate new examples of covariance functions which do not exhibit regular varying asymptotic behaviour at infinity. However, variances of averaged functionals of these fields are regularly varying. Limit theorems for weighted functionals of cyclical long-range dependent fields are obtained. The order of normalizing constants and relations between the weight functions and singularities in non-degenerative asymptotics are discussed.