Upper classes for the increments of the fractional Wiener process

Summary LetX(t) be a fractional Wiener process, i.e., a centered Gaussian process on [0, ) with stationary increments and varianceEX²(t)=t², anda(t) a positive nondecreasing function witha(t)t. We investigate the a.s. asymptotic behaviour of the incrementsI(t, a (t))=max{X{u+a(t))–X(u): 0ut–a(t)} (and some others that are similarly defined) ast.