Limit theorems for fragmentation processes with immigration

TL;DR

This paper extends limit theorems for fragmentation processes to include immigration, analyzing the asymptotic behavior of block sizes and empirical measures using submartingale techniques.

Contribution

It introduces new limit theorems for fragmentation processes with immigration, expanding understanding of their asymptotic properties.

Findings

Limit theorem for processes with immigration

Asymptotic behavior of empirical measures

Decay rate of largest block size

Abstract

In this paper we extend two limit theorems which were recently obtained for fragmentation processes to such processes with immigration. More precisely, in the setting with immigration we consider a limit theorem for the process counted with a random characteristic as well as the asymptotic behaviour of an empirical measure associated with the stopping line corresponding to the first blocks, in their respective line of descent, that are smaller than a given size. In addition, we determine the asymptotic decay rate of the size of the largest block in a homogeneous fragmentation process with immigration. The techniques used to proves these results are based on submartingale arguments.