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2007 Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks
Jean-Maxime Labarbe, Jean-Francois Marckert
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Electron. J. Probab. 12: 229-261 (2007). DOI: 10.1214/EJP.v12-397


A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being +1 or -1, equally likely. The other families quoted in the title are Bernoulli random walks under various conditions. A peak in a trajectory is a local maximum. In this paper, we condition the families of trajectories to have a given number of peaks. We show that, asymptotically, the main effect of setting the number of peaks is to change the order of magnitude of the trajectories. The counting process of the peaks, that encodes the repartition of the peaks in the trajectories, is also studied. It is shown that suitably normalized, it converges to a Brownian bridge which is independent of the limiting trajectory. Applications in terms of plane trees and parallelogram polyominoes are provided, as well as an application to the ``comparison'' between runs and Kolmogorov-Smirnov statistics.


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Jean-Maxime Labarbe. Jean-Francois Marckert. "Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks." Electron. J. Probab. 12 229 - 261, 2007.


Accepted: 11 March 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1128.60035
MathSciNet: MR2299918
Digital Object Identifier: 10.1214/EJP.v12-397

Primary: 60J65
Secondary: 60B10

Keywords: Bernoulli random walks , bridge , Brownian meander , excursion , peaks , weak convergence

Vol.12 • 2007
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