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Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks

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1.	Title	Title of document	Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks
2.	Creator	Author's name, affiliation, country	Jean-Maxime Labarbe; LMV, Université de Versailles Saint-Quentin en Yvelines
2.	Creator	Author's name, affiliation, country	Jean-Francois Marckert; LaBRI, Université Bordeaux 1
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Weak convergence; Bernoulli random walks; Brownian meander; bridge; excursion; peaks
3.	Subject	Subject classification	60J65; 60B10
4.	Description	Abstract	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2007-03-11
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/397
10.	Identifier	Digital Object Identifier	10.1214/EJP.v12-397
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 12
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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