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

7.5 STOCHASTIC CONTINUITY

In the next few sections, we examine properties of continuous-time random processes that are covered in courses on calculus regarding functions: continuity, derivatives, and integrals. Since the process is random, we might consider examining the continuity of each realization and possibly compute derivatives or integrals depending on the application. Of course, it is not feasible to perform this for all individual realizations, and even if this were possible, it is not clear that such a characterization would provide us with useful information about the collective behavior of the ensemble. As was done in previous chapters on random variables and random processes, it is preferable that we develop probabilistic formulations for these calculus considerations. DEs representing linear systems with random inputs are also discussed, which will be seen again in Chapter 8. We begin with the definition of continuity for a nonrandom function.

Definition: Continuous Function Function x(t) is continuous at to if for there exists such that

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In general, δ depends on and to which we can write as the function . This definition is consistent with our intuition about ...