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

6.7 INDEPENDENT INCREMENTS

In this section, we consider an important type of random process that can be viewed as a generalization of an iid process, and of which some well-known random processes are special cases. Let X(t) be a random process for . Since X(t) at any time instant is a random variable, the difference (increment) X(t1)−X(t2) for fixed t1 and t2 is also a random variable. This result follows from the techniques in Chapter 4 used for transformations of random variables: the pdf of X(t1)−X(t2) can be computed from those of X(t1) and X(t2).

Definition: Independent Increment Process Random process X(t) has independent increments if for any t1<t2<t3<t4:

(6.42)

This definition is a straightforward extension of independent random variables applied to disjoint increments of a random process. However, note that the increments can be contiguous, overlapping at individual time instants, such as t2 and t3 for the following:

(6.43)

The definition also applies to the situation where t1 is specified relative to the origin (starting point of the process) for which X(t0) = x0 = 0 is fixed (nonrandom). Thus

(6.44)

Specific examples of independent increment processes are provided later. ...