Convergence to Weighted Fractional Brownian Sheets

TL;DR

This paper introduces weighted fractional Brownian sheets, a new class of Gaussian fields, and demonstrates their convergence as limits of occupation time fluctuations in a particle model, offering a kernel-free approximation method.

Contribution

The paper defines weighted fractional Brownian sheets and proves their convergence as limits of occupation time fluctuations without using kernel-based representations.

Findings

Weighted fractional Brownian sheets include fractional Brownian sheets as special cases.

They can be obtained as limits of occupation time fluctuations in a stochastic particle model.

The approximation method does not rely on kernel-based integral representations.

Abstract

We define weighted fractional Brownian sheets, which are a class of Gaussian random fields with four parameters that include fractional Brownian sheets as special cases, and we give some of their properties. We show that for certain values of the parameters the weighted fractional Brownian sheets are obtained as limits in law of occupation time fluctuations of a stochastic particle model. In contrast with some known approximations of fractional Brownian sheets which use a kernel in a Volterra type integral representation of fractional Brownian motion with respect to ordinary Brownian motion, our approximation does not make use of a kernel.