An invariance principle for random walk bridges conditioned to stay positive

作者：Caravenna Francesco^*; Chaumont Loic

来源：Electronic Journal of Probability, 2013, 18(0): 1-32.

DOI：10.1214/EJP.v18-2362

• Random Walk
• Bridge
• Excursion
• Stable Law
• Levy Process
• Conditioning to Stay Positive
• Local Limit Theorem
• Invariance Principle

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 [42] and Doney [21]. We review and extend these relations to the absolutely continuous setting.

• 出版日期2013-6-5