Limiting Behavior Of Additive Functionals On The Stable Tree

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 $\max (\alpha ,\beta )\to \mathrm{\infty }$, 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.