Global Regime for General Additive Functionals of Conditioned Bienaym{\'e}-Galton-Watson Trees

TL;DR

This paper establishes an invariance principle for additive functionals of conditioned Bienaym{é}-Galton-Watson trees in the stable regime, linking them to additive functionals of stable Lévy trees, and identifies phase transitions based on toll functions.

Contribution

It extends invariance principles to a broad class of offspring distributions in the domain of attraction of stable laws, including finite variance cases, and characterizes phase transitions for toll functions.

Findings

Invariance principle for additive functionals in the stable regime

Connection to additive functionals of stable Lévy trees

Phase transition characterized by an integral test

Abstract

We give an invariance principle for very general additive functionals of conditioned Bienaym{\'e}-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable L{\'e}vy tree. This includes the case when the offspring distribution has finite variance (the L{\'e}vy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.