On the continuity 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^δ= \lim_{T\to\infty} \frac{1}{T^d} E \{\sup_{ t\in [0,T]^d \cap δ\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.

20 pages