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

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 $Q_{B_H}(t)={\sup }_{-\infty <s\le t}(B_H(t)-B_H(s)-(t-s))$, $t\ge 0$, 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 $p\ge 0$, ${\lim }_{t\to \infty }{\xi }_p(t)=\infty$ and ${\mathrm{limsup}}_{t\to \infty }({\xi }_p(t)-t)=0$ a.s. Complementary, we prove an Erdös--Révész type law of the iterated logarithm lower bound on ${\xi }_p(t)$, i.e., ${\mathrm{liminf}}_{t\to \infty }({\xi }_p(t)-t)/h_p(t)=-1$ a.s., $p>1$; ${\mathrm{liminf}}_{t\to \infty }\log ({\xi }_p(t)/t)/(h_p(t)/t)=-1$ a.s., $p\in (0,1]$, where $h_p(t)=(1/z_p(t))p\log \log t$.

arXiv admin note: text overlap with arXiv:1606.01502

References in corpus (2)

• On the Supremum of gamma-reflected Processes with Fractional Brownian Motion as Input
• An Erdös--Révész type law of the iterated logarithm for order statistics of a stationary Gaussian process

Cited by in corpus (1)

• Sample path properties of reflected Gaussian processes