Higher order expansions via Stein method

TL;DR

This paper develops a method using Stein's approach to derive higher order Edgeworth expansions for stochastic processes, demonstrating its application to Poisson and Brownian motion approximations.

Contribution

It introduces a systematic way to obtain functional Edgeworth expansions of any order via Stein's method, extending previous work.

Findings

Derived higher order expansions for Poisson and Brownian approximations.

Showed differences in the form of expansions for different stochastic processes.

Validated the method's effectiveness through specific process applications.

Abstract

This paper is a sequel of \cite{CD:2012}. We show how to establish a functional Edgeworth expansion of any order thanks to the Stein method. We apply the procedure to the Brownian approximation of compensated Poisson process and to the linear interpolation of the Brownian motion. It is then apparent that these two expansions are of rather different form.