Ball throwing on spheres

TL;DR

This paper explores the limit behavior of ball throwing on spheres, revealing a Gaussian limit that is locally self-similar for certain parameters, differing from the Euclidean case.

Contribution

It introduces the analysis of ball throwing on spheres, showing a Gaussian limit with local self-similarity, contrasting with the fractional Brownian motion limit in Euclidean spaces.

Findings

Gaussian limit on spheres

Local self-similarity for H<1/2

Different behavior from Euclidean case

Abstract

Ball throwing on Euclidean spaces has been considered for a while. A suitable renormalization leads to a fractional Brownian motion as limit object. In this paper we investigate ball throwing on spheres. A different behavior is exhibited: we still get a Gaussian limit but which is no longer a fractional Brownian motion. However the limit is locally self-similar when the self-similarity index $H$ is less than 1/2.