Rescaled weighted random balls models and stable self-similar random fields

TL;DR

This paper studies weighted random balls with heavy-tailed distributions in Euclidean space, revealing various asymptotic regimes and identifying stable self-similar random fields, including connections to known processes like fractional Brownian motion.

Contribution

It introduces a comprehensive analysis of weighted random balls models with heavy tails, uncovering multiple regimes and characterizing their statistical properties, including stable self-similar fields.

Findings

Multiple asymptotic regimes depending on parameters

Identification of stable self-similar limit fields

Connections to known processes like fractional Brownian motion

Abstract

We consider weighted random balls in $\real^d$ distributed according to a random Poisson measure with heavy-tailed intensity and study the asymptotic behaviour of the total weight of some configurations in $\real^d$. This procedure amounts to be very rich and several regimes appear in the limit, depending on the intensity of the balls, the zooming factor, the tail parameters of the radii and of the weights. Statistical properties of the limit fields are also evidenced, such as isotropy, self-similarity or dependence. One regime is of particular interest and yields $\alpha$-stable stationary isotropic self-similar generalized random fields which recovers Takenaka fields, Telecom process or fractional Brownian motion.