Fragmentation processes with an initial mass converging to infinity

Abstract

We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let $F\_{1}^{(m)}(t),F\_{2}^{(m)}(t),...$ denote the decreasing rearrangement of the masses present at time $t$ in a such process, starting from an initial mass $m$. Let then $m\to \infty $. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence $(F\_{2}^{(m)},F\_{3}^{(m)},...)$ converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of $m-F\_{1}^{(m)}$ to a stable subordinator. A continuum random tree counterpart of this result is also given: the continuum random tree describing the genealogy of a self-similar fragmentation satisfying the required assumption and starting from a mass converging to $\infty $ will converge to a tree with a spine coding a fragmentation with immigration.