Random Balls Model With Dependence

TL;DR

This paper studies a model of weighted random balls generated by a Poisson process, revealing three asymptotic regimes with stable and Poisson limits, influenced by inhomogeneity, dependence, and heavy tails.

Contribution

It introduces a new inhomogeneous dependent random balls model and characterizes its asymptotic behavior across different regimes, including stable and Poisson limits.

Findings

Three asymptotic regimes depending on parameters

Two regimes converge to stable fields

One regime converges to a Poisson integral

Abstract

In this article, we consider a configuration of weighted random balls in $\mathbb{R}^d$ generated according to a Poisson point process. The model investigated exhibits inhomogeneity, as well as dependence between the centers and the radii and heavy tails phenomena. We investigate the asymptotic behavior of the total mass of the configuration of the balls at a macroscopic level. Three different regimes appear depending on the intensity parameters and the zooming factor. Among the three limiting fields, two are stable while the third one is a Poisson integral bridging between the two stable regimes. For some particular choices of the inhomogeneity function, the limiting fields exhibit isotropy or self-similarity.