Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

6.3 CONSISTENCY AND EXTENSION

In the last two examples, the cdf of a random process was derived for specific time instants t. It is not always straightforward to characterize individual realizations of a random process as in those examples. Generally, in order to make the probabilistic description of a random process more precise, it will be necessary to define the joint probability distribution of the random variables across all subsets of time interval . We can view a random process as an extension of a random vector with each component of the vector indexed by an element of , which can have an infinity of elements and may or may not be countable. From this random vector description, a joint distribution is specified from which joint properties of the random variables are determined, such as their joint moments, independence, correlation, orthogonality, and so on.

Consider the subset of discrete time instants which need not be in increasing order, and there may be gaps in time from the original elements in . Although we normally view time as moving forward and usually without gaps (as ...