Generalized Pickands Constants
نویسندگان
چکیده
Pickands constants play an important role in the exact asymptotic of extreme values for Gaussian stochastic processes. By the generalized Pickands constant Hη we mean the limit Hη = lim T→∞ Hη(T ) T , where Hη(T ) = IE exp ( maxt∈[0,T ] (√ 2η(t)− σ2 η(t) )) and η(t) is a centered Gaussian process with stationary increments and variance function σ2 η(t). Under some mild conditions on σ2 η(t) we prove that Hη is well defined and we give a comparison criterion for the generalized Pickands constants. Moreover we prove a theorem result of Pickands for certain stationary Gaussian processes. As an application we obtain the exact asymptotic behavior of ψ(u) = IP(supt≥0 ζ(t)− ct > u) as u → ∞, where ζ(x) = ∫ x 0 Z(s) ds and Z(s) is a stationary centered Gaussian process with covariance function R(t) fulfilling some integrability conditions. 2000 Mathematics Subject Classification: 60G15 (primary), 60G70, 68M20 (secondary).
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