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Extremes

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

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

  • K. Dębicki
  • K. M. Kosiński
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.

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© 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

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