Generalized Pickands constants

Pickands constants play an important role in the exact asymptotic of extreme values for Gaussian stochastic processes. By the {it generalized Pickands constant ${cal H_{eta$ we mean the limit begin{eqnarray* {cal H_{eta= lim_{T to inftyfrac{ {cal H_{eta(T){T, end{eqnarray* where ${cal H_{eta(T)= Exp exp left(max_{t in [0,T] left(sqrt{2 eta(t)- sigma^2_{eta(t)right)right)$ and $eta(t)$ is a centered Gaussian process with stationary increments and variance function $sigma^2_{eta(t)$. Under some mild conditions on $sigma^2_{eta(t)$ we prove that ${cal H_{eta$ 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 $psi(u)=Prob(sup_{tge0zeta(t)-ct>u)$ as $utoinfty$, where $zeta(x)= int_0^x Z(s),ds$ and $Z(s)$ is a stationary centered Gaussian process with covariance function $R(t)$ fulfilling some integrability conditions.