Electronic Communications in Probability

A new proof of an old result by Pickands

J.M.P. Albin and Hyemi Choi

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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 $t\to0$. We give a new and direct proof of a result originally obtained by Pickands, on the asymptotic behaviour as $u\to\infty$ of the probability $\Pr\{\sup_{t\in[0,h]}\xi(t)>u\}$ that the process $\xi$ exceeds the level $u$. As a by-product, we obtain a new expression for Pickands constant $H_\alpha$.

Article information

Electron. Commun. Probab., Volume 15 (2010), paper no. 32, 339-345.

Accepted: 12 September 2010
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G15: Gaussian processes

Stationary Gaussian process Pickands constant extremes

This work is licensed under aCreative Commons Attribution 3.0 License.


Albin, J.M.P.; Choi, Hyemi. A new proof of an old result by Pickands. Electron. Commun. Probab. 15 (2010), paper no. 32, 339--345. doi:10.1214/ECP.v15-1566. https://projecteuclid.org/euclid.ecp/1465243975

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