Self-similar scaling limits of non-increasing Markov chains

We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from n and appropriately rescaled, converges in distribution, as n → ∞, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0. We discuss various applications to the study of random walks with a barrier, of the number of collisions in Λ-coalescents that do not descend from infinity and of non-consistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in the forthcoming paper [11]. AMS subject classifications: 60J05, 60J25, 60J75, 60G28.