Abstract

We give necessary and sufficient conditions for laws of large numbers to hold in L-2 for the empirical measure of a large class of branching Markov processes, including lambda-positive systems but also some lambda-transient ones, such as the branching Brownian motion with drift and absorption at 0. This is a significant improvement over previous results on this matter, which had only dealt so far with A-positive systems. Our approach is purely probabilistic and is based on spinal decompositions and many-to-few lemmas In addition, we characterize when the limit in question is always strictly positive on the event of survival, and use this characterization to derive a simple method for simulating (quasi-)stationary distributions.