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Extremes

, Volume 19, Issue 2, pp 273–302 | Cite as

Extremes of stationary Gaussian storage models

  • Krzysztof Dębicki
  • Peng Liu
Article

Abstract

For the stationary storage process {Q(t), t ≥ 0}, with \( Q(t)=\sup _{s\ge t}\left (X(s)-X(t)-c(s-t)^{\beta }\right ),\) where {X(t), t ≥ 0} is a centered Gaussian process with stationary increments, c > 0 and β > 0 is chosen such that Q(t) is finite a.s., we derive exact asymptotics of \(\mathbb {P}\left (\sup _{t\in [0,T_{u}]} Q(t)>u \right )\) and \(\mathbb {P}\left (\inf _{t\in [0,T_{u}]} Q(t)>u \right )\), as \(u\rightarrow \infty \). As a by-product we find conditions under which strong Piterbarg property holds.

Keywords

Storage process Gaussian process Pickands constant Strong Piterbarg property 

AMS 2000 Subject Classifications

Primary 60G15 Secondary 60G70 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Actuarial ScienceUniversity of LausanneLausanneSwitzerland

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