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On generalised Piterbarg constants. (English) Zbl 1390.60133

Summary: We investigate generalised Piterbarg constants \[ \mathcal P_{\alpha,\delta}^h=\lim\limits_{T \rightarrow \infty} \mathbb E\left\{ \sup\limits_{t\in \delta \mathbb Z\cap [0,T]}e^{\sqrt{2}B_{\alpha}(t)-| t|^{\alpha}- h(t)}\right\} \] determined in terms of a fractional Brownian motion \(B_\alpha\) with Hurst index \(\alpha/2 \in (0,1]\), the non-negative constant \(\delta\) 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 \(\mathcal P_{\alpha,\delta}^h\) and in special cases explicit formulas are obtained.

MSC:

60G15 Gaussian processes
60G70 Extreme value theory; extremal stochastic processes

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