SOME RESULTS ON INCREMENTS OF GAUSSIAN PROCESSES

Let \X(t) ; t \geqslant 0\ be a Gaussian process with stationary increments, X(0)=0 (a.s.), E X(t)=0 and \sigma^2(h)=E X(t+h)-X(t)^2=E X^2(h)=C_0 h^2 \alpha, 0< \alpha \leqslant 1 . In this paper, w \theta first prove that the Levy's theorem of the modulus of continuity of the Wiener process is also true for \X(t) ; t \geqslant 0\; i.e. \beginaligned & \lim _A \rightarrow 0 \sup _\substack0< < < \alpha \in 1 \\ \Delta-t< \alpha \frac|X(u)-X(v)|\left(2 \sigma^2(h) \log (1 / h)\right)^\frac12-\lim _A \rightarrow 0 \sup _0< t \sim 1-A \sup _0< s< h \frac|X(t+s)-X(t)|\left(2 \sigma^2(h) \log (1 / h)\right)^\frac13 \\ & -\lim _h \rightarrow 0 \sup _0< 1< 1-\mathrmA \frac|X(t+h)-X(t)|\left(2 \sigma^2(h) \log (1 / h)\right)^\frac12=1 \\ & \lim _h \rightarrow 0 \sup _0< t< 1-k \sup _0< s< a \frac|X(t+s)-X(t)|\left(2 \sigma^2(h)\left(\log \fract+hh+\log \log \frac1h\right)\right)^\frac12 \\ & =\lim _h \rightarrow 0 \sup _0< t< 1-\hbar \frac|X(t+h)-X(t)|\left(2 \sigma^2(h)\left(\log \fract+hh+\log \log \frac1h\right)\right)^\frac12=1 \endaligned Furthermore, we point out that some results on increments of the Wiener processes in 3 and 4 remain true for the increments of \X(t) ; t \geqslant 0\.