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

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.

13 pages. Changed title, improved result (Theorem 3.2), fixed typos. Final version published in ECP

References in corpus (4)

• A Local Limit Theorem for random walks conditioned to stay positive
• Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
• Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction
• A renewal theory approach to periodic copolymers with adsorption