Strictly-Stationary Processes and Ergodic Theory

Abstract

Stationary processes (with T = R¹ or ℤ) were defined in Chapter 1, section 3 as processes whose finite-dimensional distributions are invariant under translations of t. So far we have used only the invariance of the second-order moments (“wide-sense” stationarity), but in this chapter the full strength of stationarity will be needed. The main new probabilistic result will be the strong law of large numbers; through this we make contact with the interesting branch of analysis known as ergodic theory. Of course, if the strictly-stationary process has finite second moments the theory developed in Chapter 3 and Chapter 4 will apply as well, but the mathematical flavor of the present chapter is quite different from those earlier ones.