Normal convergence of non-localised geometric functionals and shot noise excursions

TL;DR

This paper develops a second order theory for geometric functionals of Poisson processes, applicable to non-stabilising shot-noise field excursions, providing variance formulas and convergence rates in the central limit theorem.

Contribution

It extends existing theories to non-stabilising functionals of spatial shot-noise fields, offering explicit variance and convergence results without requiring stabilization.

Findings

Derived exact asymptotic variance formulas.

Established central limit theorem with optimal convergence speed.

Applicable to a broad class of geometric functionals, including volume, perimeter, and Euler characteristic.

Abstract

This article presents a complete second order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results don't require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case the exact asymptotic variance is given, as well as the likely optimal speed of convergence in the central limit theorem. This goes through a general mixing-type condition that adapts nicely to both proving asymptotic normality and that variance is of volume order.