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In this manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (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 and Yakir (2014) in the 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.
