## Electronic Journal of Probability

### An invariance principle for random walk bridges conditioned to stay positive

#### 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 asuitable absolute continuity relation together with some local asymptotic estimates for random walks conditioned to stay positive, recently obtained by Vatutin and Wachtel and by Doney.We review and extend these relations to the absolutely continuous setting.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 60, 32 pp.

Dates
Accepted: 5 June 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064285

Digital Object Identifier
doi:10.1214/EJP.v18-2362

Mathematical Reviews number (MathSciNet)
MR3068391

Zentralblatt MATH identifier
1291.60090

Rights

#### Citation

Caravenna, Francesco; Chaumont, Loïc. An invariance principle for random walk bridges conditioned to stay positive. Electron. J. Probab. 18 (2013), paper no. 60, 32 pp. doi:10.1214/EJP.v18-2362. https://projecteuclid.org/euclid.ejp/1465064285

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