Ergodic Theory: Basic Results

Abstract

In this chapter definitions and basic results of ergodic theory are presented in a probabilistic setting. It must be stressed that this is a fundamental topic in probability theory. Results proved in this chapter are classical in a Markovian framework (ergodic theorems, representations of the invariant probability,...). It is nevertheless very helpful to realize that the Markov property does not really play a role to get these results: They also hold in a much more general (and natural) setting. As it will be seen in Chapter 11, the study of stationary point processes is quite elementary if a basic construction of ergodic theory is used (the “special flow” defined page 295). Since this subject is not standard in graduate courses on stochastic processes, most of the results are proved. The reference book Cornfeld et al. [13] gives a broader point of view of this domain. In the following (Ω,.ℱ, ℙ) is the probability space of reference.