On the maximum of conditioned random walks and tightness for pinning models

TL;DR

This paper establishes optimal bounds for the maximum of conditioned random walks and demonstrates tightness in pinning and wetting models, advancing understanding of their probabilistic behavior under various constraints.

Contribution

It provides the first optimal integrability results for conditioned random walk maxima and applies these to prove tightness in general pinning and wetting models.

Findings

Optimal integrability bounds for conditioned maxima.

Tightness results for pinning and wetting models.

Enhanced understanding of conditioned random walk behavior.

Abstract

We consider real random walks with finite variance. We prove an optimal integrability result for the diffusively rescaled maximum, when the walk or its bridge is conditioned to stay positive, or to avoid zero. As an application, we prove tightness under diffusive rescaling for general pinning and wetting models based on random walks.