The study of the analytical properties of random processes and their functionals, without a doubt, was and remains the relevant topic of the theory of random processes. The first result from which the study of the local properties of random processes began is Kolmogorov’s theorem on sample continuity with probability one. The classic result for Gaussian random processes is Dudley’s theorem. This paper is devoted to the study of local properties of sample paths of random processes that can be represented as a sum of squares of Gaussian random processes. Such processes are called square-Gaussian. We investigate the sufficient conditions of sample continuity with probability 1 for square-Gaussian processes based on the convergence of entropy Dudley type integrals. The estimation of the distribution of the continuity module is studied for square-Gaussian random processes. It is considered in detail an example with an estimator (correlogram) of the covariance function of a Gaussian stationary random process. The conditions on continuity of correlogram’s trajectories with probability one are found and the distribution of the continuity module is also estimated.
Analytical properties of sample paths of some stochastic processes
