Scaling limits for a random boxes model

TL;DR

This paper studies the asymptotic behavior of a model of randomly scattered rectangles with heavy-tailed dimensions, revealing different scaling regimes and their limiting random fields as the density increases and sizes shrink.

Contribution

It characterizes the scaling limits and identifies the limiting random fields for a Poisson random rectangles model with heavy-tailed edge distributions.

Findings

Different scaling regimes identified based on intensity and size parameters

Explicit descriptions of the limiting random fields

Statistical properties of the limits analyzed

Abstract

We consider random rectangles in $\mathbb{R}^2$ that are distributed according to a Poisson random measure, i.e., independently and uniformly scattered in the plane. The distributions of the length and the width of the rectangles are heavy-tailed with different parameters. We investigate the scaling behaviour of the related random fields as the intensity of the random measure grows to infinity while the mean edge lengths tend to zero. We characterise the arising scaling regimes, identify the limiting random fields and give statistical properties of these limits.