A functional Breuer-Major theorem with Poisson noise

TL;DR

This paper extends the functional Breuer-Major theorem from Gaussian to Poisson processes, utilizing spectral gap inequalities to establish tightness for stationary sequences derived from Poisson point processes.

Contribution

It introduces a novel extension of the Breuer-Major theorem to Poisson noise, broadening its applicability to non-Gaussian settings.

Findings

Established a Poisson version of the functional Breuer-Major theorem

Used $L^p$ spectral gap inequality to prove tightness

Demonstrated the theorem for stationary sequences from Poisson point processes

Abstract

We extend the functional Breuer-Major theorem for Gaussians to the Poisson case, where the stationary sequence arises from a Poisson point process. We use the $L^p$ spectral gap inequality of Poisson point process as a tool to prove tightness.