Strictly-Stationary Processes and Ergodic Theory
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.
KeywordsErgodic Theorem Stationary Markov Chain Recurrence Theorem Present Chapter Weierstrass Approximation Theorem
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