A $d$-dimensional Brownian motion as a weak limit from a one-dimensional Poisson process

TL;DR

This paper demonstrates how a single Poisson process can be used to construct a family of processes that converge in distribution to a multi-dimensional Brownian motion, providing a new perspective on stochastic process approximations.

Contribution

It introduces a novel method to derive multi-dimensional Brownian motion as a weak limit from a single Poisson process, expanding the understanding of process convergence.

Findings

Constructs a family of processes from a single Poisson process

Proves convergence in law to a $d$-dimensional Brownian motion

Provides a new approach to process approximation and convergence

Abstract

We show how from an unique standard Poisson process we can build a family of processes that converges in law to a $d$-dimensional standard Brownian motion for any $d \ge 1$.