On the speed of convergence of discrete Pickands constants to continuous ones

Abstract: In this manuscript, we address open questions raised by Dieker & Yakir (2014), who proposed a novel method of estimation of (discrete) Pickands constants $\mathcal{H}<sup>\delta_\alpha$ using a family of estimators $\xi^{\delta_\alpha(T),} T&gt;0$, 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 $$\mathcal{H}<em>\alpha⁰ - \mathcal{H}</em>\alpha<sup>\delta$$, 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 $\xi_\alpha<sup>\delta(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp{-\mathcal CT<sup>{\alpha}}$, as $T$ becomes large.