Let us consider the maximum $M(D)$ of a Dyck path $D$ chosen uniformly in the set of Dyck paths with $2n$ steps. We prove that the exponential moment of $M(D)$ normalized by the square root of $n$ is bounded in the limit of infinite $n$. This uniform bound justifies an assumption used in literature to prove certain estimates of high moments of large random matrices.
"Uniform bounds for exponential moment of maximum of a Dyck path." Electron. Commun. Probab. 14 327 - 333, 2009. https://doi.org/10.1214/ECP.v14-1486