A moment approach to the law of large numbers for supercritical branching Markov processes

We offer a new proof of the classical law of large numbers for a general class of branching Markov processes based on the asymptotic behaviour of the moments developed in \cite{bmoments, gonzalez2022erratum}. Moreover, we show that the law of the limiting random variable, that is the almost sure limit of the classical additive martingale, is completely determined by its moments.