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

7.2 STOCHASTIC CONVERGENCE

Convergence of a random sequence to a constant or a random variable can be useful in many applications such as parameter estimation (see Chapter 9). There are several kinds of convergence with varying degrees of strictness, but they all provide information about the behavior of a random sequence as . They also lead to the laws of large numbers which we discuss in this chapter.

In Chapter 2, we described almost sure and sure events for the abstract probability space . Event E is almost sure if P(E) = 1 or, alternatively, if P(Ec) = 0. Almost sure does not imply that E will occur. Event E is sure if it must happen and no other event can occur. Likewise, there is a distinction between an event E with probability zero P(E) = 0, but which could occur, and an event that can never occur: the impossible event.

FIGURE 7.1 Realization of a Poisson random process. Properties of the process include: continuity at time instant t or for the entire duration , the derivative at t, and the integral over the interval .

Example 7.1. Consider a Bernoulli random sequence X[k