Properties of Sample Functions of a Stationary Gaussian Process

R. L. Dobrushin

Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 1, pp. 132-134

Résumé

Let $\{\xi_t(\omega),-\infty<t<\infty\}$ be a separable stationary Gaussian process with a continuous correlation function. Then, the following alternative holds true: 1) either for almost all w the sample functions of the process $\xi_t(\omega)$ are continuous functions of $t$. 2) or there exists a $\beta>0$ such that for almost all $\omega$ the sample function $\xi_t(\omega)$ is such that $$\varlimsup_{t\to t_0}\xi_t(\omega)-\varliminf_{t\to t_0}\xi_t(\omega)\geq\beta$$ for any $t_0$. In the second case almost all sample functions have no points of first order discontinuities.