Abstract: For a centered self-similar Gaussian process $\{Y(t):t\in [0,\mathrm{\infty })\}$ and $R\ge 0$ we analyze the asymptotic behavior of $${\mathcal{H}}_Y^R(T)=\mathbf{E}\exp (\underset{t\in [0,T]}{sup}(\sqrt{2}Y(t)-(1+R){\sigma }_Y^2(t)))$$ as $T\to \mathrm{\infty }$. We prove that ${\mathcal{H}}_Y^R=\underset{T\to \mathrm{\infty }}{lim}{\mathcal{H}}_Y^R(T)\in (0,\mathrm{\infty })$ for $R>0$ and $${\mathcal{H}}_Y=\underset{T\to \mathrm{\infty }}{lim}\frac{{\mathcal{H}}_Y^0(T)}{T^{\gamma }}\in (0,\mathrm{\infty })$$ for suitably chosen $\gamma >0$. Additionally, we find bounds for ${\mathcal{H}}_Y^R$, $R>0$, and a surprising relation between ${\mathcal{H}}_Y$ and the classical Pickands constants.

2010 AMS Mathematics Subject Classification: Primary 60G15; Secondary 60G70.

Keywords and phrases: Gaussian process, extremes, Pickands constant, Piterbarg constant