On Generalised Piterbarg Constants

We investigate generalised Piterbarg constants P α , δ h = lim T → ∞ E sup t ∈ δℤ ∩ [ 0 , T ] e 2 B α ( t ) − | t | α − h ( t ) $$\mathcal{P}_{\alpha, \delta}^{h}=\lim\limits_{T \rightarrow \infty} \mathbb{E}\left\{ \sup\limits_{t\in \delta \mathbb{Z} \cap [0,T]} e^{\sqrt{2}B_{\alpha}(t)-|t|^{\alpha}- h(t)}\right\} $$ determined in terms of a fractional Brownian motion B α with Hurst index α/2∈(0,1], the non-negative constant δ and a continuous function h. We show that these constants, similarly to generalised Pickands constants, appear naturally in the tail asymptotic behaviour of supremum of Gaussian processes. Further, we derive several bounds for P α , δ h $\mathcal {P}_{\alpha , \delta }^{h}$ and in special cases explicit formulas are obtained.