Quantitative Central Limit Theorems for Poisson Point Processes via Localization.

Consider a Poisson point process on a metric space and a function thereof which consist of a sum of 'local' components. Examples can be statistics of spatial random graphs (total edge length, number of isolated points,...), or statistics of more complex models like the birth-growth model, which is an interacting particle system. It is known that when those functions exhibit a certain kind of 'local' behaviour, then they satisfy a Central Limit Theorem when the underlying point process grows.
In this talk, we will explore what 'local' precisely means, and how Stein's method, combined with Malliavin calculus can lead to a way of showing Berry-Esseen type convergence rates. I will talk about a joint work with Joseph Yukich, where we propose the notion of 'localization', a weak condition on score functions which is sufficient to induce a CLT.
We give Berry-Esseen type quantitative convergence rates and illustrate our results on several applications, among them the birth-growth model.
The proof builds on (and improves) the Malliavin-Stein method.