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Unformatted text preview: Chapter 5 Stationary One-Parameter Processes Section 5.1 describes the three main kinds of stationarity: strong, weak, and conditional. Section 5.2 relates stationary processes to the shift operators in- troduced in the last chapter, and to measure-preserving transforma- tions more generally. 5.1 Kinds of Stationarity Stationary processes are those which are, in some sense, the same at different times — slightly more formally, which are invariant under translation in time. There are three particularly important forms of stationarity: strong or strict, weak, and conditional. Definition 49 (Strong Stationarity) A one-parameter process is strongly sta- tionary or strictly stationary when all its finite-dimensional distributions are invariant under trnaslation of the indices. That is, for all τ ∈ T , and all J ∈ Fin( T ) , L ( X J ) = L ( X J + τ ) (5.1) Notice that when the parameter is discrete, we can get away with just checking the distributions of blocks of consecutive indices. Definition 50 (Weak Stationarity) A one-parameter process is weakly sta- tionary or second-order stationary when, for all t ∈ T , E [ X t ] = E [ X ] (5.2) and for all t, τ ∈ T , E [ X τ X τ + t ] = E [ X X t ] (5.3) 23 CHAPTER 5. STATIONARY PROCESSES 24 At this point, you should check that a weakly stationary process has time- invariant correlations. (We will say much more about this later.) You should also check that strong stationarity implies weak stationarity. It will turn out that weak and strong stationarity coincide for Gaussian processes, but not in general....
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- Spring '06
- Stationary process, Ergodic theory, iid, MeasurePreserving Transformations, al l Στ