4. Specification of Random Processes

The purpose of this chapter is to describe random processes and analyze their specification in terms of finite joint distributions for ordered time samples of the process. Particular emphasis is given to the computation of the second-order statistics of random processes, such as the autocorrelation, since these are heavily used by electrical engineers for signal analysis purposes, even though they provide only a very incomplete characterization of the stochastic behavior of processes, except in the Gaussian case. Several categories of random processes, such as Gaussian and Markov processes, which admit simple parametrizations, are introduced, as well as the class of stationary independent increments processes. This class contains the Wiener and Poisson processes, which play a major role in modeling random phenomena arising in electrical engineering systems.