  Consider the all-time maximum of a Brownian motion with negative drift.
Assume that this process is sampled at certain points in time, where the time
between two consecutive points is rendered by an Erlang distribution with mean
$1/\omega$. The family of Erlang distributions covers the range between
deterministic and exponential distributions. We show that the average
convergence rate as $\omega\to\infty$ for all such Erlangian sampled Brownian
motions is $O(\omega^{-1/2})$, and that the constant involved in $O$ ranges
from $-\zeta(1/2)/\sqrt{2\pi}$ for deterministic sampling to $1/\sqrt{2}$ for
exponential sampling. The basic ingredients of our analysis are a finite-series
expression for the expected maximum, an asymptotic expansion of
$\sum_{j=1}^{k-1}(1-\exp(2\pi i j/k))^{-s}$, $s\in\mathbb{R}$, as $k\to\infty$
using Euler-Maclaurin summation, and Fourier sampling of functions analytic in
an open set containing the closed unit disk.
