Stochastic Networks and Queues pp 279-302 | Cite as

# 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.

## Keywords

Ergodic Theory Ergodic Theorem Special Flow Discrete Dynamical System Stationary Point Process## Preview

Unable to display preview. Download preview PDF.

## References

- [Nev83]J. Neveu,
*Construction de files d’attente stationnaires*, Lecture notes in Control and Information Sciences, 60, Springer Verlag, 1983, pp. 31–41.Google Scholar - [Gar65]A.M. Garsia, A
*simple proof of E. Hopf’s maximal ergodictheorem*,Journal of Mathematics and Mechanics 14 (1965), 381382.Google Scholar - [AK42]W. Ambrose and S. Kakutani, Structure and continuity of measurable flows, Duke Mathematical Journal 9 (1942), 25–42.MathSciNetzbMATHCrossRefGoogle Scholar
- [Tot66]H. Totoki,
*Time changes of flows*, Memoirs of the Faculty of Sciences**20**(1966), no. 1, 27–55.MathSciNetzbMATHGoogle Scholar