  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.
