Quasi-stationary distributions for subcritical branching Markov chains

Consider a subcritical branching Markov chain. Let Zₙ denote the counting measure of particles of generation n. Under some conditions, we give a probabilistic proof for the existence of the Yaglom limit of (Zₙ)ₙ by the moment method, based on the spinal decomposition and the many-to-few formula. As a result, we give explicit integral representations of all quasi-stationary distributions of (Zₙ)ₙ, whose proofs are direct and probabilistic, and don't rely on Martin boundary theory.