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 ${\mathcal{H}}_{\alpha }^{\delta }$ using a family of estimators ${\xi }_{\alpha }^{\delta }(T),T>0$, where $\alpha \in (0,2]$ is the Hurst parameter, and $\delta \ge 0$ 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 $\exp \{-\mathcal{C}{T}^{\alpha }\}$, as $T$ becomes large.

21 pages

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