On laws of large numbers in L2 for supercritical branching Markov processes beyond λ-positivity

Abstract

We give necessary and sufficient conditions for laws of large numbers to hold in L2 for the empirical measure of a large class of branching Markov processes, including λ-positive systems but also some λ-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 λ-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.