Ergodic Properties

Abstract

In this chapter we formally define ergodic properties as the existence of limiting sample averages, and we study the implications of such properties. We shall see that if sample averages converge for a sufficiently large class of measurements, e.g., the indicator functions of all events, then the random process must have a property called asymptotic mean stationarity and that there is a stationary measure, called the stationary mean of the process, that has the same sample averages. In addition, it will be seen that the limiting sample averages can be interpreted as conditional probabilities or conditional expectations and that under certain conditions convergence of sample averages implies convergence of the corresponding expectations to a single expectation with respect to the stationary mean. Finally we shall define ergodicity of a process and show that it is a necessary condition for limiting sample averages to be constants instead of random variables.