An invariance principle for random walk bridges conditioned to stay positive

TL;DR

This paper establishes an invariance principle for positive-conditioned random walk bridges within the domain of attraction of stable laws, unifying discrete and continuous cases and extending known results to broader settings.

Contribution

It introduces a new invariance principle for conditioned random walk bridges, including cases with stable laws and absolutely continuous distributions, expanding previous results.

Findings

Convergence of conditioned random walk bridges to Brownian excursions.

Extension of local asymptotic estimates for positive-conditioned random walks.

Unified treatment of discrete and continuous cases in the invariance principle.

Abstract

We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite variance random walks. The proof exploits a suitable absolute continuity relation together with some local asymptotic estimates for random walks conditioned to stay positive, recently obtained by Vatutin and Wachtel [38] and Doney [21]. We review and extend these relations to the absolutely continuous setting.