Some remarks on properties of generalized Pickands constants

We study properties of it generalized Pickands constants ${cal H_{eta$, that appear in the extreme value theory of Gaussian processes and are defined via the limit begin{eqnarray* {cal H_{eta= lim_{T to inftyfrac{ {cal H_{eta(T){T, end{eqnarray* where ${cal H_{eta(T)= Exp exp left(max_{t in [0,T] left(sqrt{2 eta(t)- Var(eta(t))right)right)$ and $eta(t)$ is a centered Gaussian process with stationary increments. We give estimates of the rate of convergence of $frac{ {cal H_{eta(T){T$ to ${cal H_{eta$ and prove that if $eta_{(n)(t)$ weakly converges in $C([0,infty))$ to $eta(t)$, then under some weak conditions $lim_{ntoinfty{cal H_{eta_{(n)={cal H_{eta$. As an application we prove that $Upsilon(alpha)={cal H_{B_{alpha/2$ is continuous on $(0,2]$, where $B_{alpha/2(t)$ is a fractional Brownian motion with Hurst parameter $alpha/2$. It contradicts the conjecture that $Upsilon(alpha)$ is discontinuous at $alpha=1$.