Stationary Random Functions (Processes)

Abstract

Random processes that proceed in time with approximate homogeneity and have the form of continuous random oscillations about a certain mean value are widespread. Their probability characteristics do not depend on the choice of time reference point, i.e. are invariant relative to the shift of time. Accordingly, a random function X(t) is defined as stationary, if the probability characteristics of a random function X (t + t’) at any t’ coincide with the appropriate characteristics of X(t). This occurs only when the mathematical expectation and the variance of a random function do not depend on time, and the correlation function depends only on the difference of arguments (t’ − t). The stationary process may be considered as a process, that proceeds in time without limit In this context the stationary process is similar to the steady-state vibrations, whose parameters are independent of a time reference point.