A centra-limit theorem for conservative fragmentation chain

TL;DR

This paper establishes a central-limit theorem for a conservative fragmentation process, providing an exact rate of convergence for the empirical measure of small fragments, advancing understanding of the process's probabilistic behavior.

Contribution

It introduces a central-limit theorem for the empirical measure of fragments, offering precise convergence rates under specific assumptions, extending prior convergence and rate bounds.

Findings

Proves a central-limit theorem for the fragmentation process.

Derives an exact rate of convergence for the empirical measure.

Extends previous results on convergence bounds.

Abstract

We are interested in a fragmentation process. We observe fragments frozen when their sizes are less than {\epsilon} ({\epsilon} > 0). It is known ([BM05]) that the empirical measure of these fragments converges in law, under some renormalization. In [HK11], the authors show a bound for the rate of convergence. Here, we show a central-limit theorem, under some assumptions. This gives us an exact rate of convergence.