AMS Processes
Łukasz Dębowski · 2020
This chapter explores the general setting of ergodic theory, and considers asymptotic mean stationarity (AMS) measures and some useful operations on them. It introduces the formal definitions of AMS and pseudo-AMS probability measures. The chapter presents an operation which can be used to construct nontrivial AMS measures, called quasiperiodic coding. In the special case of switching between the process consisting of letters and the process consisting of words, a synchronizable coding corresponds to the intuitive requirement that words are separated by some delimiters such as spaces or commas. The chapter deals with the entropy rate of AMS and pseudo-AMS processes. It shows that the entropy rate of a pseudo-AMS process is less than the entropy rate of its stationary mean. In contrast, for two-sided AMS processes, not only the respective entropy rates are equal but also the process satisfies the claim of the Shannon–McMillan–Breiman theorem.