Let $D_{2n}$ be a Dyck path chosen uniformly in the set of Dyck paths with $2n$ steps. The aim of this note is to show that for any $\lambda>0$ the sequence $`E\l(\exp\l(\lambda (2n)^{-1/2}\max D_{2n}\r)\r)$ converges, and then is bounded uniformly in $n$. The uniform bound justifies an ``hypothesis'' used in the literature, for proving certain estimates of high moments of large random matrices.
Publié le : 2009-07-05
Classification:
Dyck paths,
Bernoulli bridge,
simple random walks,
random matrices,
AMS: 60C05, 60G70, 60F99,
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
@article{hal-00410386,
author = {Marckert, Jean-Fran\c cois and Khorunzhiy, Oleksiy},
title = {Uniform bounds for the exponential moments of the maximum of Dyck paths},
journal = {HAL},
volume = {2009},
number = {0},
year = {2009},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00410386}
}
Marckert, Jean-François; Khorunzhiy, Oleksiy. Uniform bounds for the exponential moments of the maximum of Dyck paths. HAL, Tome 2009 (2009) no. 0, . http://gdmltest.u-ga.fr/item/hal-00410386/