The complex Brownian motion as a strong limit of processes constructed from a Poisson process

TL;DR

This paper constructs a family of processes from a Poisson process that converges to a complex Brownian motion, providing both in-law and almost sure convergence results along with the convergence rate.

Contribution

It introduces a novel method to approximate complex Brownian motion using processes derived from a single Poisson process, including almost sure convergence analysis.

Findings

Processes converge in law to complex Brownian motion.

Almost sure convergence of the constructed processes.

Explicit rate of convergence derived.

Abstract

We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly on the unit time interval. Finally the rate of convergence is derived.