Rerooting multi-type branching trees: the infinite spine case

TL;DR

This paper investigates the local convergence of rerooted conditioned multi-type Galton--Watson trees, revealing limit objects that depend on the types recurring infinitely often along the spine, extending Aldous's work to multitype cases.

Contribution

It introduces new local convergence results for multitype Galton--Watson trees and characterizes the limit objects based on type recurrence along the spine.

Findings

Limit objects are multitype variants of Aldous's sin-tree.

Convergence results depend on types recurring infinitely often.

Extends classical results to multitype branching processes.

Abstract

We prove local convergence results of rerooted conditioned multi-type Galton--Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine.