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

Extremes of stationary Gaussian storage models

  • Krzysztof Dębicki
  • Peng LiuEmail author


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.


Storage process Gaussian process Pickands constant Strong Piterbarg property 

AMS 2000 Subject Classifications

Primary 60G15 Secondary 60G70 


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  1. Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York (2007)Google Scholar
  2. Albin, J.M.P., Samorodnitsky, G.: On overload in a storage model, with a self-similar and infinitely divisible input. Ann. Appl. Probab. 14(2), 820–844 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation, volume 27 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  4. Dȩbicki, K., Kosiński, K.M.: On the infimum attained by the reflected fractional Brownian motion. Extremes 17(3), 431–446 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dȩbicki, K.: Ruin probability for Gaussian integrated processes. Stochastic Process. Appl. 98(1), 151–174 (2002)MathSciNetCrossRefGoogle Scholar
  6. Dieker, A.B.: Extremes of Gaussian processes over an infinite horizon. Stochastic Process. Appl. 115(2), 207–248 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Hashorva, E., Ji, L., Piterbarg, V.I.: On the supremum of γ-reflected processes with fractional Brownian motion as input. Stochastic Process. Appl. 123 (11), 4111–4127 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hüsler, J., Piterbarg, V.I.: Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83(2), 257–271 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hüsler, J., Piterbarg, V.I.: Limit theorem for maximum of the storage process with fractional Brownian motion as input. Stochastic Process. Appl. 114(2), 231–250 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes, vol. 11. Springer (1983)Google Scholar
  11. Liu, P., Hashorva, E., Ji, L.: On the γ-reflected processes with fBm input. Lithuanian Math. J. 55(3), 402–412 (2015)Google Scholar
  12. Norros, I.: A storage model with self-similar input. Queueing Systems Theory Appl. 16(3-4), 387–396 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Piterbarg, V.I.: Asymptotic methods in the theory of Gaussian processes and fields, volume 148 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1996)Google Scholar
  14. Piterbarg, V.I.: Large deviations of a storage process with fractional Brownian motion as input. Extremes 4(2), 147–164 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Reich, E.: On the integrodifferential equation of Takács. I. Ann. Math. Statist. 29, 563–570 (1958)CrossRefzbMATHGoogle Scholar

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© 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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