5 research outputs found
Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks
A Bernoulli random walk is a random trajectory starting from 0 and having
i.i.d. increments, each of them being 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
Patterns in random walks and Brownian motion
We ask if it is possible to find some particular continuous paths of unit
length in linear Brownian motion. Beginning with a discrete version of the
problem, we derive the asymptotics of the expected waiting time for several
interesting patterns. These suggest corresponding results on the
existence/non-existence of continuous paths embedded in Brownian motion. With
further effort we are able to prove some of these existence and non-existence
results by various stochastic analysis arguments. A list of open problems is
presented.Comment: 31 pages, 4 figures. This paper is published at
http://link.springer.com/chapter/10.1007/978-3-319-18585-9_