On the size of the increments of nonstationary Gaussian processes

Let {X(t), t[greater-or-equal, slanted]0} be a centred nonstationary Gaussian process with EX2(t) = C0t2[alpha] for some C0 > 0, 0<[alpha]<1, and [beta]T = 1/[sigma](aT)(2(log T/aT+log log T)1/2). In this paper the a.s. asymptotic behaviour of I(T,aT[beta]T as T-->[infinity] is studied where I(T, aT) = sup{X(t')-X(t): 0[less-than-or-equals, slant]t<t'[less-than-or-equals, slant]T,t'-t[less-than-or-equals, slant]aT}. The results obtained extend work done by M. Csörgo and P. Révérz for the Wiener process.