Advertisement

Extremes

, Volume 20, Issue 4, pp 729–749 | Cite as

An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion

Open Access
Article

Abstract

Let \(B_{H}=\{B_{H}(t):t\in \mathbb R\}\) be a fractional Brownian motion with Hurst parameter H ∈ (0,1). For the stationary storage process \(Q_{B_{H}}(t)=\sup _{-\infty <s\le t}(B_{H}(t)-B_{H}(s)-(t-s))\), t ≥ 0, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function f, \( {\mathbb P(Q_{B_{H}}(t) > f(t)\, \text { i.o.})}\) equals 0 or 1. Using this criterion we find that, for a family of functions f p (t), such that \(z_{p}(t)=\mathbb P(\sup _{s\in [0,f_{p}(t)]}Q_{B_{H}}(s)>f_{p}(t))/f_{p}(t)=\mathcal C(t\log ^{1-p} t)^{-1}\), for some \(\mathcal C>0\), \({\mathbb P(Q_{B_{H}}(t) > f_{p}(t)\, \text { i.o.})= 1_{\{p\ge 0\}}}\). Consequently, with \(\xi _{p} (t) = \sup \{s:0\le s\le t, Q_{B_{H}}(s)\ge f_{p}(s)\}\), for p ≥ 0, \(\lim _{t\to \infty }\xi _{p}(t)=\infty \) and \(\limsup _{t\to \infty }(\xi _{p}(t)-t)=0\) a.s. Complementary, we prove an Erdös–Révész type law of the iterated logarithm lower bound on ξ p (t), i.e., \(\liminf _{t\to \infty }(\xi _{p}(t)-t)/h_{p}(t) = -1\) a.s., p > 1; \(\liminf _{t\to \infty }\log (\xi _{p}(t)/t)/(h_{p}(t)/t) = -1\) a.s., p ∈ (0,1], where h p (t) = (1/z p (t))p loglog t.

Keywords

Extremes of Gaussian fields Storage processes Fractional Brownian motion Law of the iterated logarithm 

AMS 2000 Subject Classifications

Primary: 60F15 60G70; Secondary: 60G22 

Notes

Acknowledgements

We are thankful to the editor and the referee for several suggestions which improved our manuscript. K. Dębicki was partially supported by National Science Centre Grant No. 2015/17/B/ST1/01102 (2016-2019). Research of K. Kosiński was conducted under scientific Grant No. 2014/12/S/ST1/00491 funded by National Science Centre.

References

  1. Asmussen, S.: Applied Probability and Queues. Springer 2nd edn (2003)Google Scholar
  2. Asmussen, S., Albrecher, H.: Ruin Probabilities. World Scientific Publishing Co. Inc., 2nd edn (2010)Google Scholar
  3. Dieker, A.B.: Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115, 207–248 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. Dębicki, K.: Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98, 151–174 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. Dębicki, K., Kosiński, K.M.: An Erdös–Révész type law of the iterated logarithm for order statistics of a stationary Gaussian process. J. Theor. Probab. doi: 10.1007/s10959-016-0710-8 (2016)
  6. Dębicki, K., Liu, P.: Extremes of stationary Gaussian storage models. Extremes 19(2), 273–302 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. Hashorva, E., Ji, L., Piterbarg, V.I.: On the supremum of γ-reflected processes with fractional Brownian motion as input. Stoch. Process. Appl. 123, 4111–4127 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. Hüsler, J., Piterbarg, V.I.: Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. Hüsler, J., Piterbarg, V.I.: On the ruin probability for physical fractional Brownian motion. Stoch. Process. Appl. 113, 315–332 (2004a)MathSciNetCrossRefMATHGoogle Scholar
  10. Hüsler, J., Piterbarg, V.I.: Limit theorem for maximum of the storage process with fractional Brownian motion as input. Stoch. Process. Appl. 114, 231–250 (2004b)MathSciNetCrossRefMATHGoogle Scholar
  11. Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin Heidelberg New York (1983)CrossRefMATHGoogle Scholar
  12. Liu, P., Hashorva, E., Ji, L.: On the γ-reflected processes with fBm input. Lith. Math. J 55(3), 402–412 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. Norros, I.: A storage model with self-similar input. Queueing Syst. 16, 387–396 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. Piterbarg, V.I.: Large deviations of a storage process with fractional Brownian motion as input. Extremes 4, 147–164 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. Qualls, C., Watanabe, H.: An asymptotic 0-1 behavior of Gaussian processes. Ann. Math. Stat. 42(6), 2029–2035 (1971)MathSciNetCrossRefMATHGoogle Scholar
  16. Shao, Q.-M.: An Erdös-Révész type law of the iterated logarithm for stationary Gaussian processes. Probab. Theory Relat. Fields 94(1), 119–133 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. Spitzer, F.: Principles of Random Walk. Van Nostrand, Princeton (1964)CrossRefMATHGoogle Scholar
  18. Watanabe, H.: An asymptotic property of Gaussian processes. Amer. Math. Soc. 148(1), 233–248 (1970)MathSciNetCrossRefMATHGoogle Scholar
  19. Zeevi, A.J., Glynn, P.W.: On the maximum workload of a queue fed by fractional Brownian motion. Ann. Appl. Probab. 10(4), 1084–1099 (2000)MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided 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.

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland

Personalised recommendations