A new proof of an old result by Pickands

J.M.P. Albin (Chalmers University of Technology, Sweden)
Hyemi Choi (Chonbuk National University, Korea)


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 $H_\alpha$.

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Pages: 339-345

Publication Date: September 12, 2010

DOI: 10.1214/ECP.v15-1566


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