A new proof of an old result by Pickands
Hyemi Choi (Chonbuk National University, Korea)
Abstract
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
References
- Albin, J.M.P. On extremal theory for stationary processes. Ann. Probab. 18 (1990), 92-128. MR1043939 (91e:60117)
- Berman, S.M. Sojourns and extremes of stationary processes. Ann. Probab. 10 (1982), 1-46. MR0637375 (84j:60043)
- Burnecki, K. and Michna, Z. Simulation of Pickands constants. Probab. Math. Statist. 22 (2002), 193-199. MR1944151
- Leadbetter, M.R., Lindgren, G. and Rootzén, H. Extremes and related properties of random sequences and processes. Springer, New York. (1983). MR0691492 (84h:60050)
- Michina, Z. Remarks on Pickands theorem. arXiv:0904.3832v1 [math.PR]. (2009). Math. Review number not available.
- Pickands, J.III. Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145 (1969), 75-86. MR0250368 (40 #3607)
- Piterbarg, V.I. Asymptotic methods in the theory of Gaussian processes and fields. Translations of Mathematical Monographs, 148 (1996) American Mathematical Society, MR1361884 (97d:60044)
- Qualls, C. and Watanabe, H. Asymptotic properties of Gaussian processes. Ann. Math. Statist. 43 (1972), 580-596. MR0307318 (46 #6438)
- Samorodnitsky, G. Probability tails of Gaussian extrema. Stochastic Process. Appl. 38 (1991), 55-84 MR1116304 (92g:60056)
- Talagrand, M. Small tails for the supremum of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 307-315. MR0953122 (89g:60143)
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