Ergodic Properties

Abstract

Random processes and dynamical systems possess ergodic properties if the time averages of measurements converge. This chapter looks at the implications of ergodic properties and limiting sample averages. The relation between expectations of the averages and averages of the expectations are derived, and it is seen that if a process has sufficient ergodic properties, then the process must be asymptotic mean stationary and that the the properties describe a stationary measure with the same ergodic properties as the asymptotically mean stationary process. A variety of related characterizations of the system transformation are given that relate to the asymptotic behavior, including ergodicity, recurrence, mixing, block stationarity and ergodicity, and total ergodicity. The chapter concludes with the ergodic decomposition which shows that the stationary measure implied by convergent sample averages is equivalent to a mixture of stationary and ergodic processes. The ergodic theorems of the next chapter provide conditions under which a process or system will possess ergodic properties for large classes of measurements. In particular, it is shown that asymptotic mean stationarity is a sufficient condition as well as a necessary condition.