ON SOME CONTINUITY AND DIFFERENTIABILITY PROPERTIES OF PATHS OF GAUSSIAN PROCESSESl

This paper considers some path pro~erties of real separable Gaussian processes ~ with parameter set an arbitrary interval. The following results are established, among others. At every fixed point the paths of ~ are continuous, or differentiable, with probability zero or one. If ~ is measurable, then with probability one its paths have essentially the same points of continuity and differentiability. If ~ is measurable and not mean square continuous or diffe~entiable at every pc~_nt, then with probability one its paths are almost nowhere continuous or differentiable respectively. If ~ is mean square contir.uous and 8tationary, then its paths are differentiable with probability or.e if and only if ~ is mean square differentiable. If ~ is harmcnizable, then its paths are absolutely continuous if and only if ~ is mean square differentiable. Also a class of harmonizable processes is determined for which the following are true: (i) with probability one paths are either continuous or unbounded on every jnterval, and (ii) path differentiability with pr0bability one is equivalent to mean square differentiability. ON SOME CONTINUITY AND DIFFERENTIABILITY PROPERTIES OF PATHS OF GAUSSIAN PROCESSES I Stamatis Cambanis University of North Carolina at Chapel Hill