On the size of the increments of nonstationary Gaussian processes (Q797900)

Let \((X(t))_{t\geq o}\) be a centred Gaussian process with stationary increments satisfying \(X(0)=0\) and \(EX^ 2(h)=C_ oh^{2\alpha}\) for some \(0<\alpha <1\) and \(C_ o>0\), and let \(a_ T\) be an increasing function of \(T>0\) such that \(T^{-1}a_ T\) is non-increasing. Results of the form \[ \lim \sup_{T\to\infty }\beta_ T\sup_{0\leq t<t'\leq T,t'-t\leq a_ T}| X(t')-X(t)| =1 \] are established, for suitably chosen \(\beta_ T\). In the case of the Wiener process \((C_ o=1\), \(\alpha =1/2)\), such results were proved by \textit{M. Csörgö} and \textit{P. Révész} [Ann. Probab. 7, 731-737 (1979; Zbl 0412.60038)], and the present paper uses an approach similar to theirs.