Strong Law of Large Numbers for Fragmentation Processes

TL;DR

This paper establishes a strong law of large numbers for self-similar fragmentation processes, demonstrating almost sure convergence of empirical measures related to the sizes of fragments.

Contribution

It extends classical results to fragmentation processes, proving almost sure convergence for empirical measures in self-similar fragmentation models.

Findings

Almost sure convergence of empirical measures for fragmentation processes

Extension of classical laws to self-similar fragmentation

Results applicable to homogenous and self-similar processes

Abstract

In the spirit of a classical results for Crump-Mode-Jagers processes, we prove a strong law of large numbers for homogenous fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than $\eta$ for $1\geq \eta >0$.