A strong law of large numbers for branching processes: almost sure spine events

TL;DR

This paper proves a new strong law of large numbers for branching processes, showing that events almost surely occurring for the spine imply convergence of sums over the population, using measure theory and spine theory.

Contribution

It introduces a novel strong law of large numbers for branching processes with a simple proof leveraging measure-theoretic and spine techniques.

Findings

Almost sure convergence of sums over particles

Events occurring for the spine imply population convergence

Simplified proof via measure-theoretic manipulations

Abstract

We demonstrate a novel strong law of large numbers for branching processes, with a simple proof via measure-theoretic manipulations and spine theory. Roughly speaking, any sequence of events that eventually occurs almost surely for the spine entails the almost sure convergence of a certain sum over particles in the population.