On the continuity of Pickands constants

TL;DR

This paper investigates the continuity properties of Pickands constants associated with certain random fields, extending classical results and providing approximation methods for these constants as the discretization parameter tends to zero.

Contribution

The paper extends classical results on Pickands constants by establishing their continuity at zero and providing approximation methods for these constants in a broader class of random fields.

Findings

H_Z^δ is finite for δ ≥ 0 under mild assumptions.

H_Z^0 can be approximated by H_Z^δ as δ approaches zero.

Continuity of H_Z^δ at δ=0 shown for specific extensions of Pickands constants.

Abstract

For a non-negative separable random field $Z(t), t\in \mathbb{R}^d$ satisfying some mild assumptions we show that \begin{eqnarray*} H_Z^\delta = \lim_{T\to\infty} \frac{1}{T^d} E \{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } Z(t) \} <\infty \end{eqnarray*} for $\delta \ge 0$ where $0 \mathbb{Z}^d := \mathbb{R}^d$ and prove that $H_Z^0$ can be approximated by $H_Z^\delta$ if $\delta$ tends to 0. These results extend the classical findings for the Pickands constants $H_{Z}^\delta$, defined for $Z(t)= \exp\left( \sqrt{ 2} B_\alpha (t)- |t|^{2\alpha }\right), t\in \mathbb{R}$ with $B_\alpha$ a standard fractional Brownian motion with Hurst parameter $\alpha \in (0,1]$. The continuity of $H_{Z}^\delta$ at $\delta=0$ is additionally shown for two particular extensions of Pickands constants.