Abstract: For a centered self-similar Gaussian process \(\{Y(t):t\in[0,\infty)\}\) and \(R\ge0\) we analyze the asymptotic behavior of \[\mathcal{H}_Y^R(T) = \mathbf{E} \exp \left( \sup_{t \in [0,T]} \bigl(\sqrt{2}\, Y(t) - (1+R) \sigma_Y^2(t) \bigr)\right)\] as \(T\to\infty\). We prove that \(\mathcal{H}_Y^R=\lim_{T\to\infty} \mathcal{H}_Y^R(T)\in(0,\infty)\) for \(R>0\) and \[\mathcal{H}_Y=\lim_{T\to\infty} \frac{\mathcal{H}_Y^0(T)}{T^\gamma}\in(0,\infty)\] 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