A new proof of an old result by Pickands. (English) Zbl 1227.60068

Albin, J. M. P.; Choi, Hyemi

Electron. Commun. Probab. 15, 339-345 (2010).

Summary: Let \(\{\xi(t)\}_{t\in [0,h]}\) be a stationary Gaussian process with covariance function \(r\) such that \(r(t)=1-C|t|^\alpha+o(|t|^\alpha)\) as \(t\to 0\). We give a new and direct proof of a result originally obtained by J. Pickands, III [Trans. Am. Math. Soc. 145, 75–86 (1969; Zbl 0206.18901)], on the asymptotic behaviour as \(u\to\infty\) of the probability P\(\{\sup_{t \in [0,h]} \xi(t)>u\}\) that the process \(\xi\) exceeds the level \(u\). As a by-product, we obtain a new expression for Pickands constant \(H_\alpha\).

MSC:

60G70	Extreme value theory; extremal stochastic processes
60G15	Gaussian processes
60G10	Stationary stochastic processes

Keywords:

stationary Gaussian process; Pickands constant; extremes

Citations:

Zbl 0206.18901