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

TL;DR

This work analyzes the speed at which discrete Pickands constants $\mathcal{H}_{\alpha}^{\delta}$ converge to the continuous constants $\mathcal{H}_{\alpha}$ as the discretization step $\delta$ tends to zero, addressing questions raised by Dieker and Yakir. Using the Dieker-Yakir representation $\mathcal{H}_{\alpha}^{\delta}=\mathbb{E}\{\xi_{\alpha}^{\delta}\}$ and Gaussian-process techniques centered on $Z_{\alpha}(t)=\sqrt{2}B_{\alpha}(t)-|t|^{\alpha}$, the authors derive precise upper bounds for the discretization error $\mathcal{H}_{\alpha}-\mathcal{H}_{\alpha}^{\delta}$: for $\alpha\in(0,1)$ the rate is $\mathcal{O}(\delta^{\alpha/2})$, while for $\alpha\in(1,2)$ it is $\mathcal{O}(\delta^{\alpha/2}|\log\delta|^{1/2})$. They obtain exact asymptotics for the boundary cases $\alpha=1$ and $\alpha=2$, with $\lim_{\delta\to0}(\mathcal{H}_{1}-\mathcal{H}_{1}^{\delta})/\sqrt{\delta} = -\zeta(1/2)/\sqrt{\pi}$ and $\lim_{\delta\to0}(\mathcal{H}_{2}-\mathcal{H}_{2}^{\delta})/\delta^{2} = 1/(12\sqrt{\pi})$, and provide explicit formulas for $\mathcal{H}_{1}^{\delta}$ and $\mathcal{H}_{2}^{\delta}$. In addition, the paper proves uniform moment bounds for $\xi_{\alpha}^{\delta}(T)$ and shows the truncation bias decays at least as $e^{-\mathcal{C}T^{\alpha}}$, yielding stable sampling behavior. These results substantiate the conjectured discretization rates and offer practical guidance for Monte Carlo estimation of Pickands constants via discretized fBm.

Abstract

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

Theorems & Definitions (31)

• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• proof : Proof of Theorem \ref{['thm:main']}, $\alpha\in (0,1)$.
• Lemma 3.4