On the large increments of fractional Brownian motion (Q1273005)

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.