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.
"On the maximum of conditioned random walks and tightness for pinning models." Electron. Commun. Probab. 23 1 - 13, 2018. https://doi.org/10.1214/18-ECP172