Self-similar scaling limits of non-increasing Markov chains

Abstract

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 our paper, Scaling limits of Markov branching trees, with applications to Galton–Watson and random unordered trees (2010).