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Invariance Principle for the Random Walk Conditioned to Have Few Zeros

  • Laurent Serlet
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)

Abstract

We consider a nearest neighbor random walk on \(\mathbb{Z}\) starting at zero, conditioned to return at zero at time 2n and to have a number z n of zeros on (0, 2n]. As \(n \rightarrow +\infty \), if \(z_{n} = o(\sqrt{n})\), we show that the rescaled random walk converges toward the Brownian excursion normalized to have unit duration. This generalizes the classical result for the case z n  ≡ 1.

Keywords

Conditioned random walk Excursions Invariance principle 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques (CNRS umr 6620)Clermont Université, Université Blaise PascalAubièreFrance

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