Scaling limits of a heavy tailed Markov renewal process

TL;DR

This paper studies heavy tailed Markov renewal processes, proving their convergence to a stable regenerative set, and applies these results to analyze the scaling limit of a constrained random walk model at criticality.

Contribution

It establishes the convergence of heavy tailed Markov renewal processes to a stable regenerative set and applies this to the strip wetting model at criticality.

Findings

Heavy tailed Markov renewal processes converge to a $eta$-stable regenerative set.

The results characterize the scaling limit of the strip wetting model at criticality.

Provides a framework for analyzing constrained random walks with heavy tails.

Abstract

In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $\ga$-stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0,\infty) \times [0,a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.