Zooming in at the root of the stable tree

Abstract

We study the shape of the normalized stable Lévy tree $\mathcal{T}$ 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

$${\mathbf{Z}}_{\mathit{\alpha },\mathit{\beta }}={\int }_{\mathcal{T}}\mathrm{\mu }(\mathrm{d}x){\int }_0^{H(x)}{\mathit{\sigma }}_{r,x}^{\mathit{\alpha }}{\mathfrak{h}}_{r,x}^{\mathit{\beta }}\mathrm{d}r$$

as $max(\mathit{\alpha },\mathit{\beta })\to \mathrm{\infty }$, where μ is the mass measure on $\mathcal{T}$, $H(x)$ is the height of x and ${\mathit{\sigma }}_{r,x}$ (resp. ${\mathfrak{h}}_{r,x}$) is the mass (resp. height) of the subtree of $\mathcal{T}$ 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.