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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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)
Arendarczyk, M., Dȩbicki, K.: Exact asymptotics of supremum of a stationary Gaussian process over a random interval. Stat. Prob. Lett. 82, 645–652 (2012)
Dȩbicki, K., Rolski, T.: A note on transient Gaussian fluid models. Queueing Systems 41, 321–342 (2002)
Dȩbicki, K., van Uitert, M.: Large buffer asymptotics for generalized processor sharing queues with Gaussian inputs. Queueing Syst. 54, 111–120 (2006)
Foss, S., Korshunov, D., Zachary, S.: An introduction to Heavy-tailed and Subexponential Distributions, 2nd edn.Springer-Verlag, New York (2013)
Fotopoulos, S., Luo, Y.: Subordinated Gaussian Processes, the Log-Return Principles. Washington State University, Washington (2011)
Geman, H., Madan, D.B., Yor, M.: Time changes for Lévy processes. Math. Financ. 11, 79–96 (2001)
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)
Hüsler, J., Piterbarg, V.: Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83, 257–271 (1999)
Kozubowski, T.J., Meerschaert, M.M., Molz, F. J., Lu, S.: Fractional Laplace model for hydraulic conductivity. Geophhys. Res. Lett. 31, 1–4 (2004)
Kozubowski, T.J., Meerschaert, M.M., Podgórski, K.: Fractional Laplace motion. Adv. Appl. Probab. 38, 451–464 (2006)
Michna, Z.: Self-similar processes in collective risk theory. J. Appl. Math. Stoch. Anal. 11, 429–448 (1998)
Palmowski, Z., Zwart, B.: Tail asymptotics of the supremum of a regenerative process. J. Appl. Prob. 44, 349–365 (2007)
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)
Wu, R., Wang, W.: The hitting time for a Cox risk process. J. Comp. Appl. Math 236, 2706–2716 (2012)
Zwart, B., Borst, S., Dȩbicki, K.: Subexponential asymptotics of hybrid fluid and ruin models. Ann. Appl. Probab. 15, 500–517 (2005)
Author information
Authors and Affiliations
Corresponding author
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.
About this article
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
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