A moment approach for the convergence of spatial branching processes to the Continuum Random Tree

TL;DR

This paper proves that certain critical spatial branching processes with type-dependent reproduction converge to the Brownian Continuum Random Tree, using a novel moment-based approach and a new many-to-few formula.

Contribution

It introduces a general method for establishing invariance principles for branching processes by analyzing their moments through a new many-to-few formula.

Findings

Tree structures converge to the Brownian Continuum Random Tree

Moment convergence is achieved via a new many-to-few formula

Applicable to a broad class of type-dependent branching processes

Abstract

We consider a general class of branching processes in discrete time, where particles have types belonging to a Polish space and reproduce independently according to their type. If the process is critical and the mean distribution of types converges for large times, we prove that the tree structure of the process converges to the Brownian Continuum Random Tree, under a moment assumption. We provide a general approach to prove similar invariance principles for branching processes, which relies on deducing the convergence of the genealogy from computing its moments. These are obtained using a new many-to-few formula, which provides an expression for the moments of order $k$ of a branching process in terms of a Markov chain indexed by a uniform tree with $k$ leaves.