Generalized Random Processes

TL;DR: The characteristic functional of a generalized random process generalizes the notion of the characteristic function of a probability distribution and enables one to introduce processes with continuously varying time whose values at distinct times are independent random variables.

Abstract: This chapter discusses generalized random processes. The normative concept of a random function is based upon the assumption that it is possible to measure the value of the random function at every moment of time t without calculating the value of the function at other moments of time. As a result of the smoothing action of apparatuses, one can, thus, obtain a probability distribution not only for processes that exist at each instant of time t but also for generalized processes for which there do not exist probability distributions at isolated instants of time. The characteristic functional of a generalized random process generalizes the notion of the characteristic function of a probability distribution. Applying the theory of generalized random processes enables one to introduce processes with continuously varying time whose values at distinct times are independent random variables. In doing this, it is possible to establish a connection between processes with independent values at every point and infinitely divisible random variables. Also, it is always convenient to carry out the study of processes with independent values at every point with the help of their characteristic functionals. The generalized random functions of several variables can also be referred to as functions generalized random fields. Subsequently, a substantial portion of the theory of generalized random fields is analogous to the corresponding portion of the theory of generalized random processes. Finally, the analog of the notion of a stationary generalized random process is that of a homogeneous generalized random field.