Chapter 3 - Random quantities, processes, and fields

Abstract

This chapter discusses the basic properties of random quantities, processes, and fields that are widely used in analyzing dynamic systems with fluctuating parameters. It is found that if one deals with random function, then all this function statistical characteristics at any fixed instant are exhaustively described in terms of the one-point probability density. The Fourier transform of the correlation function with respect to the spatial variable defines the spatial spectral function. The typical realization curve of random process concerns the fundamental features of the behavior of a separate process realization as a whole for temporal intervals of arbitrary duration. The one-point probability density of random process is a result of averaging the singular indicator function over an ensemble of realizations of this process. The discontinuous processes are the random functions that change their time-dependent behavior at discrete instants given statistically. The characteristic functional that describes all statistical characteristics of random process is analyzed. The one-dimensional discrete-continuous Markovian process is also elaborated.