Asymptotics for the small fragments of the fragmentation at nodes

We consider the fragmentation at nodes of the Lévy continuous random tree introduced in a previous paper. In this framework we compute the asymptotic behaviour of the number of small fragments at time θ. This limit is increasing in θ and discontinuous. In the α-stable case the fragmentation is self-similar with index l /α, with α ∈ (1, 2), and the results are close to those Bertoin obtained for general self-similar fragmentations but with an additional assumption which is not fulfilled here.