Weak approximation for Gaussian processes from renewal processes

TL;DR

This paper demonstrates how renewal processes can be used to approximate Gaussian processes like fractional Brownian motion, providing conditions for convergence and expanding the toolkit for stochastic process approximation.

Contribution

It introduces new methods to approximate Gaussian processes using renewal processes and establishes conditions for convergence, including fractional Brownian motion.

Findings

Renewal processes can approximate fractional Brownian motion.

Sufficient conditions for convergence of renewal-based approximations.

Application to multiple Stratonovich integrals.

Abstract

In previous works, Bardina and Rovira (2023) constructed a family of processes that converge strongly towards Brownian motion, defined from renewal processes, are constructed. In this paper we prove that some of these processes can be utilized to build approximations of Gaussian processes such as fractional Brownian motion or multiple Stratonovich integrals and we provide sufficient conditions on renewal processes to ensure that the convergence holds. An illustrative example of such a Gaussian process is the fractional Brownian motion with any Hurst parameter.