A natural modification of a random process and its application to stochastic functional series and Gaussian measures

Abstract

It is known that a Gaussian stochastic process can be expanded in a functional series with random independent coefficients. In the case where the process is continuous in mean but there exists no modification of it with continuous simple functions, the series does not converge uniformly. In what cases does it converge pointwise? This question reduces to the well-studied problem of the boundedness of the sample functions. It is shown that the pointwise convergence of the expansion mentioned above is equivalent to the continuity of the sample functions of the process in a certain separable metric. Some other properties of Gaussian processes and measures are considered, ansd generalizations to the non-Gaussian case are given.