Chapter 4 - Random quantities, processes and fields

Abstract

This chapter discusses the basic concepts of the theory of random quantities, processes, and fields. It discusses the concept of typical realization curve of random process, which concerns the fundamental features of the behavior of a separate process realization as a whole for temporal intervals of arbitrary duration. Consideration of specific random processes allows the obtaining an additional information concerning the realization's spikes relative to the typical realization curve. 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. This function is concentrated at points at which the process crosses the line. The chapter starts the discussion with continuous processes, namely, with the Gaussian random process with zero-valued mean and correlation function. The chapter also considers the random processes whose points of discontinuity form Poisson streams of points. Currently, three types of such processes–the Poisson process, telegrapher's process, and generalized telegrapher's process–are mainly used in model problems of physics.