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.
Highlights
Invariance principles for conditioned random walks have a long history, going back at least to the work of Liggett [33], who proved that the bridge of a random walk in the domain of attraction of a stable law, suitably rescaled, converges in distribution toward the bridge of the corresponding stable Lévy process
Iglehart [31], Bolthausen [9] and Doney [19] focused on a different type of conditioning: they proved invariance principles for random walks conditioned to stay positive over a finite time interval, obtaining as a limit the analogous conditioning for the corresponding Lévy process, known as meander
Given a random walk in the domain of attraction of a stable law, we show that its bridge conditioned to stay positive, suitably rescaled, converges in distribution toward the bridge of the corresponding stable Lévy process conditioned to stay positive
Summary
Invariance principles for conditioned random walks have a long history, going back at least to the work of Liggett [33], who proved that the bridge of a random walk in the domain of attraction of a stable law, suitably rescaled, converges in distribution toward the bridge of the corresponding stable Lévy process. Iglehart [31], Bolthausen [9] and Doney [19] focused on a different type of conditioning: they proved invariance principles for random walks conditioned to stay positive over a finite time interval, obtaining as a limit the analogous conditioning for the corresponding Lévy process, known as meander. Some more technical details are deferred to the appendices
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