Ergodic Properties of Markov Processes
transition and contraction operators; ratio ergodic theorem; space-time invariance and tail triviality; mixing and convergence in total variation; Harris recurrence and transience; existence and uniqueness of invariant measure; distributional and pathwise limits
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- The first ratio ergodic theorems were obtained by Doeblin (1938b), Doob (1938,1948a), Kakutani (1940), and Hurewicz (1944). Hopf (1954) and Dunford and Schwartz (1956) extended the pointwise ergodic theorem to general L 1 -L 8-contractions, and the ratio ergodic theorem was extended to positive. L 1-contractions by Chacon and Ornstein (1960). The present approach to their result in due to Akcoglu and Chacon (1970).Google Scholar
- The notion of Harris recurrence goes back to Doeblin (1940) and Harris (1956). The latter author used the condition to ensure the existence, in discrete time, of a σ-finite invariant measure. A corresponding continuous-time result was obtained by H. Watanabe (1964). The total variation convergence of Markov transition probabilities was obtained for a countable state space by Orey (1959, 1962) and in general by Jamison and Orey (1967). Blackwell and Freedman (1964) noted the equivalence of mixing and tail triviality. The present coupling approach goes back to Griffeath (1975) and S. Goldstein (1979) for the case of strong ergodicity and to Berbee (1979) and Aldous and Thorisson (1993) for the corresponding weak result.Google Scholar
- There is an extensive literature on ergodic theorems for Markov processes, mostly dealing with the discrete-time case. General expositions have been given by many authors, beginning with Neveu (1971) and Orey (1971). Our treatment of Harris recurrent Feller processes is adapted from Kunita (1990), who in turn follows the discrete-time approach of Revuz (1984). Krengel (1985) gives a comprehensive survey of abstract ergodic theorems. Detailed accounts of the coupling method and its various ramifications appear in Lindvall (1992) and Thorisson (2000).Google Scholar