On the genealogy on conditioned stable L\'evy forest

TL;DR

This paper constructs a conditioned stable Lévy forest from unconditioned forests and proves an invariance principle showing convergence of associated processes to a stable Lévy process and its height process as the size grows.

Contribution

It provides a new realization of conditioned stable Lévy forests and establishes their convergence to continuous limits via an invariance principle.

Findings

Convergence of coding random walk, contour, and height processes to stable Lévy process and height process.

New realization of conditioned stable Lévy forests from unconditioned forests.

Asymptotic behavior of Galton-Watson trees conditioned on total progeny.

Abstract

We give a realization of the stable L\'evy forest of a given size conditioned by its mass from the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering $k$ independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to $n$. We prove that when $n$ and $k$ tend towards $+\infty$, under suitable rescaling, the associated coding random walk, the contour and height processes converge in law on the Skorokhod space respectively towards the "first passage bridge" of a stable L\'evy process with no negative jumps and its height process.