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A note on upper estimates for Pickands constants. (English) Zbl 1283.60068

Summary: Pickands constants \(\mathcal{H}_{B_\alpha}\) play a significant role in the extreme value theory of Gaussian processes. Recall that
\[ \mathcal{H}_{B_\alpha}:=\lim_{T\to\infty} \frac {\displaystyle\mathbb{E}\exp\left(\sup_{t\in[0,T]} (\sqrt{2} B_{\alpha}(t) - t^{\alpha}) \right)}{T}, \]
where \(\{B_\alpha(t), t\geq0\}\) is a fractional Brownian motion with Hurst parameter \(\alpha/2\) and \(\alpha \in(0,2]\).
In this note we derive new upper bounds for \(\mathcal{H}_{B_\alpha}\) and \(\alpha\in(1,2]\). The obtained results improve bounds given by Q.-M. Shao [Stat. Sin. 6, No. 1, 245–257 (1996; Zbl 0841.60036)].

MSC:

60G15 Gaussian processes
60G70 Extreme value theory; extremal stochastic processes
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems

Citations:

Zbl 0841.60036
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References:

[1] Bräker, K., 1993. High boundary excursions of locally stationary Gaussian processes. Ph.D. Thesis. University of Bern, Bern; Bräker, K., 1993. High boundary excursions of locally stationary Gaussian processes. Ph.D. Thesis. University of Bern, Bern
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