# An Erd\\"os-R\\'ev\\'esz type law of the iterated logarithm for reflected fractional Brownian motion

Source: [An Erd\\"os-R\\'ev\\'esz type law of the iterated logarithm for reflected fractional Brownian motion](<https://www.sota2.com/research/paper/68fa547c1e209131ce98c3ff>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-27T05:06:49.965Z

Authors: K. D\\k{e}bicki, K.M. Kosi\\'nski

Published: 2016-12-29T18:07:58.000Z

## Abstract

Let $B\_H=\\{B\_H(t):t\\in\\mathbb R\\}$ be a fractional Brownian motion with Hurst parameter $H\\in(0,1)$. For the stationary storage process $Q\_{B\_H}(t)=\\sup\_{-\\infty\<s\\le t}(B\_H(t)-B\_H(s)-(t-s))$, $t\\ge0$, 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)=\\mathscr C(t\\log^{1-p} t)^{-1}$, for some $\\mathscr C\>0$, $\\mathbb P(Q\_{B\_H}(t) \> f\_p(t)\\, \\text{ i.o.})= 1\_{\\{p\\ge 0\\}}$. Consequently, with $\\xi\_p (t) = \\sup\\{s:0\\le s\\le t, Q\_{B\_H}(s)\\ge f\_p(s)\\}$, for $p\\ge 0$, $\\lim\_{t\\to\\infty}\\xi\_p(t)=\\infty$ and $\\limsup\_{t\\to\\infty}(\\xi\_p(t)-t)=0$ a.s. Complementary, we prove an Erd\\"os--R\\'ev\\'esz type law of the iterated logarithm lower bound on $\\xi\_p(t)$, 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\\in(0,1\]$, where $h\_p(t)=(1/z\_p(t))p\\log\\log t$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1612.09229>)

[PDF](<https://arxiv.org/pdf/1612.09229.pdf>)