Convergence of weighted branching processes

TL;DR

This paper investigates the long-term behavior of complex weighted branching processes, extending classical probabilistic results to more general settings with infinite types and non-standard rescaling, with applications to random trees and convergence metrics.

Contribution

It introduces new theoretical results for weighted multi-type branching processes, including laws of large numbers and martingale convergence in broader contexts.

Findings

Extended classical laws of large numbers to infinite-type settings

Proved convergence in Wasserstein distance for mean semi-groups

Established ergodic average convergence along lineages

Abstract

We study the long-term behavior of weighted multi-type branching processes, focusing on extending classical laws of large numbers and martingale convergence to settings with infinitely many weighted particles, arbitrary type spaces and non-geometric rescaling. We demonstrate applications to Galton-Watson trees indexed by random weights and by random kernels, convergence in Wasserstein distance of the underlying mean semi-group, and convergence of ergodic averages along lineages.