Discrete approximation of stable white noise - Application to spatial linear filtering

TL;DR

This paper develops methods for discretely approximating stable white noise, enabling simulation of stable random fields and spatial processes, with proven convergence and applications to moving average fields and Brownian Lévy motion.

Contribution

It introduces grid and shot noise approximation techniques for stable white noise, with rigorous convergence theorems and applications to spatial filtering and Lévy motion.

Findings

Proved convergence of discrete approximations to stable white noise.

Applied results to spatial linear filtering and heavy-tailed random fields.

Extended approximation methods to Brownian Lévy motion on spheres and Euclidean spaces.

Abstract

Motivated by the simulation of stable random fields, we consider the issue of discrete approximations of independently scattered stable noise. Two approaches are proposed: grid approximations available when the underlying space is $\bbR^d$ and shot noise approximations available on more general spaces. Limit theorems stating the convergence of discrete random noises to stable white noise are proved. These results are then applied to study moving average spatial random fields with heavy-tailed innovations and related limit theorems. A second application deals with discrete approximation for Brownian L\'evy motion on the sphere or on the euclidean space.