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

TL;DR

This paper investigates how quickly discrete Pickands constants converge to their continuous counterparts, providing bounds on discretization error and analyzing the properties of estimators for these constants.

Contribution

We derive an upper bound for the discretization error of Pickands constants and analyze the convergence rate of estimators, confirming conjectures for certain parameter ranges.

Findings

Discretization error bounds match conjectured rates for lpha in (0,1].

All moments of the estimator are uniformly bounded.

Bias of the estimator decays exponentially with T.

Abstract

In this manuscript, we address open questions raised by Dieker \& Yakir (2014), who proposed a novel method of estimation of (discrete) Pickands constants $\mathcal{H}^\delta_\alpha$ 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.