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.
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References
Asmussen, S.: Applied Probability and Queues. Springer 2nd edn (2003)
Asmussen, S., Albrecher, H.: Ruin Probabilities. World Scientific Publishing Co. Inc., 2nd edn (2010)
Dieker, A.B.: Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115, 207–248 (2005)
Dębicki, K.: Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98, 151–174 (2002)
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)
Dębicki, K., Liu, P.: Extremes of stationary Gaussian storage models. Extremes 19(2), 273–302 (2016)
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)
Hüsler, J., Piterbarg, V.I.: Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271 (1999)
Hüsler, J., Piterbarg, V.I.: On the ruin probability for physical fractional Brownian motion. Stoch. Process. Appl. 113, 315–332 (2004a)
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)
Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin Heidelberg New York (1983)
Liu, P., Hashorva, E., Ji, L.: On the γ-reflected processes with fBm input. Lith. Math. J 55(3), 402–412 (2015)
Norros, I.: A storage model with self-similar input. Queueing Syst. 16, 387–396 (2004)
Piterbarg, V.I.: Large deviations of a storage process with fractional Brownian motion as input. Extremes 4, 147–164 (2001)
Qualls, C., Watanabe, H.: An asymptotic 0-1 behavior of Gaussian processes. Ann. Math. Stat. 42(6), 2029–2035 (1971)
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)
Spitzer, F.: Principles of Random Walk. Van Nostrand, Princeton (1964)
Watanabe, H.: An asymptotic property of Gaussian processes. Amer. Math. Soc. 148(1), 233–248 (1970)
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)
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.
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Dębicki, K., Kosiński, K.M. An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion. Extremes 20, 729–749 (2017). https://doi.org/10.1007/s10687-017-0296-2
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DOI: https://doi.org/10.1007/s10687-017-0296-2