A moment approach for the convergence of spatial branching processes to the Continuum Random Tree

We consider a general class of branching processes in discrete time, where particles have types belonging to a Polish space and reproduce independently according to their type. If the process is critical and the mean distribution of types converges for large times, we prove that the tree structure of the process converges to the Brownian Continuum Random Tree, under a moment assumption. We provide a general approach to prove similar invariance principles for branching processes, which relies on deducing the convergence of the genealogy from computing its moments. These are obtained using a new many-to-few formula, which provides an expression for the moments of order $k$ of a branching process in terms of a Markov chain indexed by a uniform tree with $k$ leaves.