# Limit theorems for filtered long-range dependent random fields

## TL;DR

This paper explores the limit distributions of functionals of filtered long-range dependent Gaussian random fields, revealing non-Gaussian limits and the influence of observation window shapes on the Hurst parameter.

## Contribution

It extends existing results by establishing convergence for Hurst parameters in (0, 0.5) and analyzing the effect of observation window shapes on the Hurst parameter.

## Key findings

- Limit distributions are non-Gaussian.
- Convergence established for H in (0, 0.5).
- Hurst parameter depends on observation window shape.

## Abstract

This article investigates general scaling settings and limit distributions of functionals of filtered random fields. The filters are defined by the convolution of non-random kernels with functions of Gaussian random fields. The case of long-range dependent fields and increasing observation windows is studied. The obtained limit random processes are non-Gaussian. Most known results on this topic give asymptotic processes that always exhibit non-negative auto-correlation structures and have the self-similar parameter $H\in(\frac{1}{2},1)$. In this work we also obtain convergence for the case $H\in(0,\frac{1}{2})$ and show how the Hurst parameter $H$ can depend on the shape of the observation windows. Various examples are presented.