2006/01/25 by Jean-Maxime Labarbe, Labarbe, Jean-Maxime, Jean‐François Marckert +2 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Diffusion and Search Dynamics #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #cs.DM #math.PR
paper · pdf · doi:10.48550/arxiv.math/0601624
arxiv created 2006/01/25 · openalex publication_date 2006/01/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 cited in the title are Bernoulli random walks under various conditionings. 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 also provided.