Self-similar scaling limits of non-increasing Markov chains

TL;DR

This paper investigates the scaling limits of non-increasing Markov chains, showing they converge to self-similar processes under certain conditions, with applications to various stochastic models.

Contribution

It introduces the convergence of rescaled non-increasing Markov chains to self-similar processes, extending understanding of their asymptotic behavior.

Findings

Rescaled chains converge to self-similar Markov processes.

Joint convergence of absorption times and chain states.

Applications to random walks, coalescents, and branching trees.

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 $n\rightarrow \infty$, 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 $\Lambda$-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).