  In this manuscript, we address open questions raised by Dieker \& Yakir
(2014), who proposed a novel method of estimation of (discrete) Pickands
constants $\mathcal{H}^\delta_\alpha$ using a family of estimators
$\xi^\delta_\alpha(T), T>0$, where $\alpha\in(0,2]$ is the Hurst parameter, and
$\delta\geq0$ is the step-size of the regular discretization grid. We derive an
upper bound for the discretization error $\mathcal{H}_\alpha^0 -
\mathcal{H}_\alpha^\delta$, whose rate of convergence agrees with Conjecture 1
of Dieker & Yakir (2014) in case $\alpha\in(0,1]$ and agrees up to logarithmic
terms for $\alpha\in(1,2)$. Moreover, we show that all moments of
$\xi_\alpha^\delta(T)$ are uniformly bounded and the bias of the estimator
decays no slower than $\exp\{-\mathcal CT^{\alpha}\}$, as $T$ becomes large.
