Stated in simple terms, we may say:

A stochastic process is a set of random variables indexed in time.

Elaborating on this succinct statement, we find that in many of the real-life phenomena encountered in practice, *time* features prominently in their description. Moreover, their actual behavior has a random appearance. Referring back to the example of wireless communications briefly described in Section 3.1, we find that the received signal at the wireless channel output varies randomly with time. Processes of this kind are said to be *random* or *stochastic*;^{1} hereafter, we will use the term “stochastic.” Although probability theory does not involve time, the study of stochastic processes naturally builds on probability theory.

The way to think about the relationship between probability theory and stochastic processes is as follows. When we consider the statistical characterization of a stochastic process at a particular instant of time, we are basically dealing with the characterization of a *random variable* sampled (i.e., observed) at that instant of time. When, however, we consider a single realization of the process, we have a *random waveform* that evolves across time. The study of stochastic processes, therefore, embodies two approaches: one based on *ensemble averaging* and the other based on *temporal averaging*. Both approaches and their characterizations are considered in this chapter.

Although it is not possible to predict the ...

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