Extremal behavior of stationary marked point processes

TL;DR

This paper studies the extremal behavior of stationary marked point processes, establishing convergence to a Poisson cluster process and enabling explicit calculations of extremal quantities.

Contribution

It introduces a framework for analyzing local limits of extreme scores in marked point processes, extending existing results to allow explicit extremal calculations.

Findings

Rescaled positions and scores converge to a Poisson cluster process.

The local limit determines global asymptotics for extreme scores.

Explicit extremal quantities can be calculated from the limiting process.

Abstract

We consider stationary configurations of points in Euclidean space which are marked by positive random variables called scores. The scores are allowed to depend on the relative positions of other points and outside sources of randomness. Such models have been thoroughly studied in stochastic geometry, e.g.\ in the context of random tessellations or random geometric graphs. It turns out that in a neighbourhood of a point with an extreme score one can often rescale positions and scores of nearby points to obtain a limiting point process, which we call the tail configuration. Under some assumptions on dependence between scores, this local limit determines the global asymptotics for extreme scores within increasing windows in $\R^d$. The main result establishes the convergence of rescaled positions and clusters of high scores to a Poisson cluster process, quantifying the idea of the Poisson clumping heuristic by D.~Aldous (in the point process setting). In contrast to the existing results, our framework allows for explicit calculation of essentially all extremal quantities related to the limiting behavior of extremes. We apply our results to models based on (marked) Poisson processes where the scores depend on the distance to the $k$th nearest neighbor and where scores are allowed to propagate through a random network of points depending on their locations.