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Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Enkelejd Hashorva () and Jürg Hüsler ()
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Enkelejd Hashorva: University of Bern
Jürg Hüsler: University of Bern

Methodology and Computing in Applied Probability, 2000, vol. 2, issue 3, 255-269

Abstract: Abstract Let X(t), t∈[0,1], be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique fixed point t0 in the interval [0,1], the behavior of P{supt∈[0,1] X(t)>u} is known for u→∞. We investigate the case where the unique point t0 = tu depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes Xu(t) depending on u, which have for each u such a unique point tu tending to the boundary as u→∞. We derive the asymptotic behavior of P{supt∈[0,1] X(t)>u}, depending on the rate as tu tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the difficulties to simulate such rare events, when u is large.

Keywords: extreme values; Gaussian process; nonconstant variance; unique maximum variance point; crossing of a boundary; Ornstein-Uhlenbeck process (search for similar items in EconPapers)
Date: 2000
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DOI: 10.1023/A:1010029228490

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