Continuous Time Random Processes

Abstract

This chapter presents elements of the general theory of continuous time processes. Section 18.1 introduces the key concepts of random processes, sample paths, cylinder sets and finite-dimensional distributions, the spaces of continuous functions and functions without discontinuities of the second kind, and equivalence of random processes. Section 18.2 presents the fundamental results on regularity of processes: Kolmogorov’s theorem on existence of a continuous modification and Kolmogorov–Chentsov’s theorem on existence of an equivalent process with trajectories without discontinuities of the second kind. The section also contains discussions of the notions of separability, stochastic continuity and continuity in mean.