Upper classes for the increments of the fractional Wiener process (Q1263874)

Let X(t) be a fractional Wiener process, i.e., a centered Gaussian process on [0,\(\infty)\) with stationary increments and variance E \(X^ 2(t)=t^{2\alpha}\), and a(t) a positive nondecreasing function with a(t)\(\leq t\). We investigate the a.s. asymptotic behaviour of the increments \[ I(t,a(t))=\max \{X(u+a(t))-X(u):\quad 0\leq t-a(t)\} \] (and some others that are similarly defined as \(t\to \infty\).