Limit theorems for weighted functionals of cyclical long-range dependent random fields

TL;DR

This paper establishes limit theorems for weighted functionals of isotropic long-range dependent random fields with spectral singularities, revealing the impact of these singularities on the asymptotic behavior of various functionals.

Contribution

It provides new limit theorems showing how spectral singularities influence the asymptotics of weighted functionals, including the Donsker scheme and more general schemes.

Findings

Limit theorems are established for weighted functionals of isotropic long-range dependent fields.

For the Donsker scheme, singularities at non-zero frequencies do not affect the limit.

For general schemes, singularities at non-zero frequencies influence the asymptotic behavior.

Abstract

The paper investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of functionals, which includes the Donsker scheme, the limit is not affected by singularities at non-zero frequencies. For general schemes, in contrast to the Donsker line, we demonstrate that the singularities at non-zero frequencies play a role even for linear functionals.