Poisson hulls

TL;DR

This paper develops a new estimation framework for linear and higher order statistics of Poisson point processes using a hull operator, with applications to convex body volume and integral estimation, leveraging a spatial Markov property.

Contribution

It introduces a hull operator-based estimation scheme for Poisson processes, providing error analysis and convergence rates, including for convex bodies and H"older functions.

Findings

Estimation error characterized by Kabanov–Skorohod integral.

Derived rate of normal convergence for estimators.

Applicable to convex body volume and integral estimation.

Abstract

We introduce a hull operator on Poisson point processes, the easiest example being the convex hull of the support of a point process in Euclidean space. Assuming that the intensity measure of the process is known on the set generated by the hull operator, we discuss estimation of an expected linear statistic built on the Poisson process. In special cases, our general scheme yields an estimator of the volume of a convex body or an estimator of an integral of a H\"older function. We show that the estimation error is given by the Kabanov--Skorohod integral with respect to the underlying Poisson process. A crucial ingredient of our approach is a spatial strong Markov property of the underlying Poisson process with respect to the hull. We derive the rate of normal convergence for the estimation error, and illustrate it on an application to estimators of integrals of a H\"older function. We also discuss estimation of higher order symmetric statistics.