Stein approximation for multidimensional Poisson random measures by third cumulant expansions

TL;DR

This paper develops Stein approximation bounds for multidimensional Poisson stochastic integrals using third cumulant Edgeworth expansions, potentially achieving faster convergence than traditional methods.

Contribution

It introduces a novel approach leveraging third cumulant expansions and Malliavin calculus to improve convergence rates in Poisson approximation.

Findings

Third cumulant expansions can accelerate convergence rates.

Stein bounds are established for Poisson stochastic integrals.

Method applies to multidimensional Poisson measures.

Abstract

We obtain Stein approximation bounds for stochastic integrals with respect to a Poisson random measure over ${\Bbb R}^d$, $d\geq 2$. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard Berry-Esseen rate for some sequences of Poisson stochastic integrals.