Sub-exponential tail bounds for conditioned stable Bienaym\'e-Galton-Watson trees

TL;DR

This paper derives uniform sub-exponential tail bounds for key structural properties of large conditioned Bienaymé-Galton-Watson trees with offspring distributions in the stable law domain, extending finite variance results.

Contribution

It extends tail bound results for critical Bienaymé-Galton-Watson trees to the stable law domain, covering width, height, and outdegree.

Findings

Established sub-exponential tail bounds for tree properties

Extended finite variance results to stable law cases

Applicable to large conditioned trees with stable offspring distributions

Abstract

We establish uniform sub-exponential tail bounds for the width, height and maximal outdegree of critical Bienaym\'e-Galton-Watson trees conditioned on having a large fixed size, whose offspring distribution belongs to the domain of attraction of a stable law. This extends results obtained for the height and width by Addario-Berry, Devroye & Janson in the finite variance case.