Weak convergence towards two independent Gaussian processes from a unique Poisson process

TL;DR

This paper demonstrates how two independent Gaussian processes can be approximated by processes derived from a single Poisson process, with applications to fractional and sub-fractional Brownian motions.

Contribution

It introduces a novel construction of processes from a single Poisson process that converge to two independent Gaussian processes, including fractional Brownian motion.

Findings

Finite-dimensional distributions converge in law to the Gaussian processes.

Constructed processes from a single Poisson process can approximate fractional Brownian motion.

Method provides a new way to simulate Gaussian processes using Poisson processes.

Abstract

We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As an application of this result we obtain families of processes that converge in law towards fractional Brownian motion and sub-fractional Brownian motion.