Limit theorems for fragmentation processes with immigration

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 the asymptotic behaviour of an empirical measure associated with the stopping line corresponding to the first blocks, in their respective line of descent, of size less than η ∈ (0, 1] as well as a limit theorem for the process counted with a random characteristic. 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.