  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.
