# On Generalised Piterbarg Constants

Long Bai (), Krzysztof Dȩbicki (), Enkelejd Hashorva () and Li Luo ()
Long Bai: University of Lausanne, UNIL-Dorigny
Krzysztof Dȩbicki: University of Wrocław
Enkelejd Hashorva: University of Lausanne, UNIL-Dorigny
Li Luo: University of Lausanne, UNIL-Dorigny

Methodology and Computing in Applied Probability, 2018, vol. 20, issue 1, 137-164

Abstract: Abstract 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.

Keywords: Pickands constants; Piterbarg constants; Gaussian process; Extremes; Exact asymptotics; Brown-Resnick stationarity; 60G15; 60G70 (search for similar items in EconPapers)
Date: 2018
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