Pickands-Piterbarg constants for self-similar Gaussian processes

Abstract

For a centered self-similar Gaussian process $\{Y(t):t\in [0,\infty )\}$ and $R\ge 0$ we analyze asymptotic behaviour of \[ \mathcal{H}_Y^R(T) \; = \; \mathbf{E} \exp \left( \sup_{t \in [0,T]} \sqrt{2} Y(t) - (1+R) σ_Y^2(t) \right), \] 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 \[\mathcal{H}_Y=\lim_{T\to\infty} \frac{\mathcal{H}_Y^0(T)}{T^γ}\in(0,\infty)\] 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 classical Pickands constants.

19 pages