关于Gauss过程增量的若干结果

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SOME RESULTS ON INCREMENTS OF GAUSSIAN PROCESSES

Abstract: 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. $$ \begin{aligned} & \lim _{A \rightarrow 0} \sup _{\substack{0< < < \alpha \in 1 \\ \Delta-t< \alpha}} \frac{|X(u)-X(v)|}{\left(2 \sigma^2(h) \log (1 / h)\right)^{\frac{1}{2}}}-\lim _{A \rightarrow 0} \sup _{0< t \sim 1-A} \sup _{0< s< h} \frac{|X(t+s)-X(t)|}{\left(2 \sigma^2(h) \log (1 / h)\right)^{\frac{1}{3}}} \\ & -\lim _{h \rightarrow 0} \sup _{0< 1< 1-\mathrm{A}} \frac{|X(t+h)-X(t)|}{\left(2 \sigma^2(h) \log (1 / h)\right)^{\frac{1}{2}}}=1 \\ & \lim _{h \rightarrow 0} \sup _{0< t< 1-k} \sup _{0< s< a} \frac{|X(t+s)-X(t)|}{\left(2 \sigma^2(h)\left(\log \frac{t+h}{h}+\log \log \frac{1}{h}\right)\right)^{\frac{1}{2}}} \\ & =\lim _{h \rightarrow 0} \sup _{0< t< 1-\hbar} \frac{|X(t+h)-X(t)|}{\left(2 \sigma^2(h)\left(\log \frac{t+h}{h}+\log \log \frac{1}{h}\right)\right)^{\frac{1}{2}}}=1 \end{aligned} $$ Furthermore, we point out that some results on increments of the Wiener processes in [3] and [4] remain true for the increments of $\{X(t) ; t \geqslant 0\}$.