On Generalised Piterbarg Constants

  • Long Bai
  • Krzysztof Dȩbicki
  • Enkelejd Hashorva
  • Li Luo
Article
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Abstract

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 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 \(\mathcal {P}_{\alpha , \delta }^{h}\) and in special cases explicit formulas are obtained.

Keywords

Pickands constants Piterbarg constants Gaussian process Extremes Exact asymptotics Brown-Resnick stationarity 

Mathematics Subject Classification (2010)

60G15 60G70 
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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Actuarial ScienceUniversity of Lausanne, UNIL-DorignyLausanneSwitzerland
  2. 2.Mathematical InstituteUniversity of WrocławWrocławPoland

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