Limiting Behavior Of Additive Functionals On The Stable Tree

TL;DR

This paper investigates the local structure of stable Lévy trees near their root, revealing a universal limit described by a Poisson point process, and studies the asymptotic behavior of certain additive functionals as parameters grow large.

Contribution

It characterizes the limiting shape of stable Lévy trees near the root and analyzes the asymptotics of complex additive functionals related to these trees.

Findings

The local limit of the stable Lévy tree near the root is the unnormalized Kesten tree.

The limit is described by a tree-valued Poisson point process independent of normalization.

As parameters grow, additive functionals exhibit specific asymptotic behaviors.

Abstract

We study the shape of the normalized stable L\'{e}vy 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 \[\mathbf{Z}_{\alpha,\beta}=\int_{\mathcal{T}} \mu(\mathrm{d} x) \int_0^{H(x)} \sigma_{r,x}^\alpha \mathfrak{h}_{r,x}^\beta\,\mathrm{d} r\]as $\max(\alpha,\beta) \to \infty$, where $\mu$ is the mass measure on $\mathcal{T}$, $H(x)$ is the height of $x$ and $\sigma_{r,x}$ (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 Bienaym{\'e}-Galton-Watson trees.