Scaling limit of multitype Galton-Watson trees with infinitely many types

TL;DR

This paper establishes a scaling limit for multitype Galton-Watson trees with infinitely many types, showing convergence to reflected Brownian motion and extending previous finite-type results to countably infinite types.

Contribution

It introduces a new class of 2-type Galton-Watson trees with edge lengths and proves a convergence result, extending invariance principles to infinitely many types.

Findings

Weighted height functions converge to reflected Brownian motion

Invariance principle extended to countably infinite types

Results generalize finite-type Galton-Watson tree behavior

Abstract

We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton--Watson trees with a countable number of types, thus extending a result of G. Miermont on multitype Galton--Watson trees with finitely many types.