Self-similar scaling limits of Markov chains on the positive integers

TL;DR

This paper investigates the asymptotic behavior of certain Markov chains on positive integers, establishing a limit theorem that links their scaled behavior to self-similar Markov processes and identifying different regimes of recurrence.

Contribution

It extends previous work by analyzing Markov chains with rare large jumps and small steps, providing a comprehensive limit theorem and applications to various stochastic processes.

Findings

Limit theorem linking scaled Markov chains to self-similar processes

Identification of three regimes: transient, recurrent, positive-recurrent

Applications to Bessel-type walks and coalescence-fragmentation processes

Abstract

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If $X_{n}$ is such a Markov chain started at $n$, we establish a limit theorem for $\frac{1}{n}X_{n}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by $X_{n}$ to reach some fixed finite set. We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which $X_{n}$ exhibits different behavior. The present results extend those of Haas & Miermont who focused on the case of non-increasing Markov chains. We further present a number of applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks, nonnegative self-similar Markov processes, invariance principles for random walks conditioned to stay positive, and exchangeable coalescence-fragmentation processes.