Skip to main content

Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals

Abstract

Let {X(t),t ≥ 0} be a centered Gaussian process and let γ be a non-negative constant. In this paper we study the asymptotics of \(\mathbb {P} \left \{\underset {t\in [0,\mathcal {T}/u^{\gamma }]}\sup X(t)>u\right \}\) as \(u\rightarrow \infty \), with \(\mathcal {T}\) an independent of X non-negative random variable. As an application, we derive the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes.

References

  • Adler, R.J.: An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, Inst. Math. Statist. Lecture Notes Monogr. Ser. 12, Inst. Math. Statist, Hayward, CA (1990)

  • Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer (2007)

  • Arendarczyk, M., Dȩbicki, K.: Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. Bernoulli 17, 194–210 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Arendarczyk, M., Dȩbicki, K.: Exact asymptotics of supremum of a stationary Gaussian process over a random interval. Stat. Prob. Lett. 82, 645–652 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Dȩbicki, K., Rolski, T.: A note on transient Gaussian fluid models. Queueing Systems 41, 321–342 (2002)

    Article  MathSciNet  Google Scholar 

  • Dȩbicki, K., van Uitert, M.: Large buffer asymptotics for generalized processor sharing queues with Gaussian inputs. Queueing Syst. 54, 111–120 (2006)

    Article  MathSciNet  Google Scholar 

  • Foss, S., Korshunov, D., Zachary, S.: An introduction to Heavy-tailed and Subexponential Distributions, 2nd edn.Springer-Verlag, New York (2013)

    Book  Google Scholar 

  • Fotopoulos, S., Luo, Y.: Subordinated Gaussian Processes, the Log-Return Principles. Washington State University, Washington (2011)

    Google Scholar 

  • Geman, H., Madan, D.B., Yor, M.: Time changes for Lévy processes. Math. Financ. 11, 79–96 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Hashorva, E., Ji, L., Piterbarg, V.I.: On the supremum of gamma-reflected processes with fractional Brownian motion as input. Stoch. Proc. Appl. 123, 4111–4127 (2013)

    Article  MathSciNet  Google Scholar 

  • Hüsler, J., Piterbarg, V.: Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83, 257–271 (1999)

    MATH  Google Scholar 

  • Kozubowski, T.J., Meerschaert, M.M., Molz, F. J., Lu, S.: Fractional Laplace model for hydraulic conductivity. Geophhys. Res. Lett. 31, 1–4 (2004)

    Google Scholar 

  • Kozubowski, T.J., Meerschaert, M.M., Podgórski, K.: Fractional Laplace motion. Adv. Appl. Probab. 38, 451–464 (2006)

    Article  MATH  Google Scholar 

  • Michna, Z.: Self-similar processes in collective risk theory. J. Appl. Math. Stoch. Anal. 11, 429–448 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Palmowski, Z., Zwart, B.: Tail asymptotics of the supremum of a regenerative process. J. Appl. Prob. 44, 349–365 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Piterbarg, V.I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields. In: Transl. Math. Monogr., vol. 148. AMS, Providence (1996)

  • Resnick, S.I: Extreme Values (1987). Regular Variation, and Point Processes. Springer

  • Tan, Z., Hashorva, E.: Exact tail asymptotics of the supremum of strongly dependent gaussian processes over a random interval. Lith. Math. J 53, 91–102 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  • Wu, R., Wang, W.: The hitting time for a Cox risk process. J. Comp. Appl. Math 236, 2706–2716 (2012)

    Article  MATH  Google Scholar 

  • Zwart, B., Borst, S., Dȩbicki, K.: Subexponential asymptotics of hybrid fluid and ruin models. Ann. Appl. Probab. 15, 500–517 (2005)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Krzysztof Dębicki.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dębicki, K., Hashorva, E. & Ji, L. Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals. Extremes 17, 411–429 (2014). https://doi.org/10.1007/s10687-014-0186-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10687-014-0186-9

Keywords

  • Tail asymptotics
  • Large deviations
  • Weibullian tails
  • Supremum over random intervals
  • Gaussian process
  • Fractional Brownian motion
  • Fractional laplace motion
  • Gamma process
  • Ruin probability

AMS 2000 Subject Classifications

  • Primary 60G15
  • Secondary 60G70
  • 60K30
  • 91B30