On the large increments of fractional Brownian motion
type: Item
instance of: scholarly article
title: On the large increments of fractional Brownian motion
zbMATH Open document ID: 0919.60045
DOI: 10.1016/S0167-7152(98)00140-0
author: Charles El-Nouty
published in: Statistics \& Probability Letters
publication date: 1999-08-31T00:00:00Z
review text: Let \(\{B_H(t), t\geq 0\}\) be a fractional Brownian motion with index \(0< H<1\), i.e., a centered Gaussian process with stationary increments satisfying \(B_H(0)= 0\), with probability 1, and \(E(B_H(t))^2= t^{2H}\), \(t\geq 0\). Let \(a_T\) be a nondecreasing function of \(T\geq 0\) such that \(0\leq a_T\leq T\); \(a_T/T\) is a nonincreasing function of \(T\geq 0\), such that \(\lim_{T\to\infty} (\ln_2T)^{-1}\ln T/a_T= r\in [0,\infty]\), where \(\ln u= \log(u\vee e)\) and \(\ln_2u= \ln(\ln u)\) for \(u\geq 0\). Consider the process \(\{V_T, T\geq 0\}\) defined by \[ V_T= \sup_{0\leq s\leq T-a_T}| B_H(s+ a_T)- B_H(s)|\left/ a^H_T(2(\ln(T/a_T)+\right. \ln_2T))^{1/2}. \] We set \(\tau= \lim_{T\to\infty} a_T/T\). If \(\tau>0\), then \(\liminf_{T\to\infty} V_T= 0\) a.s. If \(\tau= 0\) and \(0< H\leq 1/2\) or \(\tau= 0\), \(1/2< H<1\) and \(r> 4^H/(4- 4^H)\), then \(\liminf_{T\to\infty} V_T= (r/(r+ 1))^{1/2}\) a.s.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 1228685
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: Slépian's lemma
describes a project that uses: longmemo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1228685