Zooming in at the root of the stable tree

Abstract

We study the shape of the normalized stable Lévy tree T near its root. We show that, when zooming in at the root at the proper speed with a scaling depending on the index of stability, we get the unnormalized Kesten tree. In particular the limit is described by a tree-valued Poisson point process which does not depend on the initial normalization. We apply this to study the asymptotic behavior of additive functionals of the form Zα,β= ∫Tμ(dx)∫0H(x)σ r,xαh r,xβdr as max(α,β)→∞, where μ is the mass measure on T, H(x) is the height of x and σr,x (resp. hr,x) is the mass (resp. height) of the subtree of T above level r containing x. Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaymé-Galton-Watson trees.