On the continuity of Pickands constants

@article{Dbicki2022OnTC,
  title={On the continuity of Pickands constants},
  author={Krzysztof Dȩbicki and Enkelejd Hashorva and Zbigniew Michna},
  journal={Journal of Applied Probability},
  year={2022},
  volume={59},
  pages={187 - 201}
}
Abstract For a non-negative separable random field Z(t), $t\in \mathbb{R}^d$ , satisfying some mild assumptions, we show that $ H_Z^\delta =\lim_{{T} \to \infty} ({1}/{T^d}) \mathbb{E}\{{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } Z(t) }\} <\infty$ 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 Pickands constants $H_{Z}^\delta… 

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