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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 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.
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Dates and versions

hal-00691962 , version 1 (27-04-2012)
hal-00691962 , version 2 (09-10-2012)

Identifiers

  • HAL Id : hal-00691962 , version 2

Cite

Francesco Caravenna, Loïc Chaumont. An invariance principle for random walk bridges conditioned to stay positive. 2012. ⟨hal-00691962v2⟩
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