Strong approximations of Brownian sheet by uniform transport processes

TL;DR

This paper extends previous work on uniform transport processes approximating Brownian motion to multi-parameter cases, constructing processes from Poisson processes that converge almost surely to the Brownian sheet and higher-dimensional Wiener processes.

Contribution

It introduces a method to approximate multi-parameter Brownian sheets using uniform transport processes derived from Poisson processes, extending prior one-parameter results.

Findings

Processes constructed from Poisson processes converge almost surely to the Brownian sheet.

Extension of approximation techniques to multi-parameter Wiener processes.

Uniform convergence on the unit square and higher dimensions achieved.

Abstract

Many years ago, Griego, Heath and Ruiz-Moncayo proved that it is possible to define realizations of a sequence of uniform transform processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In this paper we extend their results to the multi parameter case. We begin constructing a family of processes, starting from a set of independent standard Poisson processes, that has realizations that converge almost surely to the Brownian sheet, uniformly on the unit square. At the end the extension to the $d$-parameter Wiener processes is presented.