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Unformatted text preview: Chapter 24 The Almost-Sure Ergodic Theorem This chapter proves Birkhoff’s ergodic theorem, on the almost- sure convergence of time averages to expectations, under the as- sumption that the dynamics are asymptotically mean stationary. This is not the usual proof of the ergodic theorem, as you will find in e.g. Kallenberg. Rather, it uses the AMS machinery developed in the last lecture, following Gray (1988, sec. 7.2), in turn following Katznelson and Weiss (1982). The central idea is that of “blocking”: break the infinite sequence up into non- overlapping blocks, show that each block is well-behaved, and conclude that the whole sequence is too. This is a very common technique in modern ergodic theory, especially among information theorists. In pure probability theory, the usual proof of the ergodic theorem uses a result called the “maximal ergodic lemma”, which is clever but somewhat obscure, and doesn’t seem to generalize well to non-stationary processes: see Kallenberg, ch. 10. We saw, at the end of the last chapter, that if time-averages converge in the long run, they converge on conditional expectations. Our work here is showing that they (almost always) converge. We’ll do this by showing that their lim infs and lim sups are (almost always) equal. This calls for some preliminary results about the upper and lower limits of time-averages. Definition 294 For any observable f , define the lower and upper limits of its time averages as, respectively, A f ( x ) ≡ lim inf t →∞ A t f ( x ) (24.1) Af ( x ) ≡ lim sup t →∞ A t f ( x ) (24.2) Define L f as the set of x where the limits coincide: L f ≡ x A f ( x ) = Af ( x ) (24.3) 161 CHAPTER 24. THE ALMOST-SURE ERGODIC THEOREM 162 Lemma 295 A f and Af are invariant functions. Proof: Use our favorite trick, and write A t f ( T x ) = t +1 t A t +1 f ( x )- f ( x ) /t . Clearly, the lim sup and lim inf of this expression will equal the lim sup and lim inf of A t +1 f ( x ), which is the same as that of A t f ( x ). Lemma 296 The set of L f is invariant. Proof: Since A f and Af are both invariant, they are both measurable with respect to I (Lemma 262), so the set of x such that A f ( x ) = Af ( x ) is in I , therefore it is invariant (Definition 261)....
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This note was uploaded on 12/20/2011 for the course STAT 36-754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
- Spring '06