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.
"A strong law of large numbers for branching processes: almost sure spine events." Electron. Commun. Probab. 19 1 - 6, 2014. https://doi.org/10.1214/ECP.v19-2641