Stationary Processes and Ergodic Theory

  • Olav Kallenberg
Part of the Probability and Its Applications book series (PIA)


Stationarity, invariance, and ergodicity; discrete- and continuous-time ergodic theorems; moment and maximum inequalities; multivariate ergodic theorems; sample intensity of a random measure; subadditivity and products of random matrices; conditioning and ergodic decomposition; shift coupling and the invariant σ-field


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  1. The history of ergodic theory dates back to Boltzmann’s (1887) work in statistical mechanics. Boltzmann’s ergodic hypothesis—the conjectural equality between time and ensemble averages—was long accepted as a heuristic principle. In probabilistic terms it amounts to the convergence (math), where X t represents the state of the system (typically the configuration of all molecules in a gas) at time t, and the expected value is computed with respect to a suitably invariant probability measure on a compact submanifold of the state space.Google Scholar
  2. The ergodic hypothesis was sensationally proved as a mathematical theorem, first in an Inversion by Von Neumann (1932), after Koopman (1931) had noted the connection between measure-preserving transformations and unitary operators on a Hilbert space, and shortly afterwards in the pointwise form of Birkhoff (1932). The initially quite intricate proof of the latter was simplified in stages: first by Yosida and Kakutani (1939), who noted how the result follows easily from the maximal ergodic Lemma 10.7, and then by Garsia (1965), who gave a short proof of the latter result. Khinchin (1933, 1934) pioneered a translation of the results of ergodic theory into the probabilistic setting of stationary sequences and processes.Google Scholar
  3. The first multivariate ergodic theorem was obtained by Wiener (1939), who proved Theorem 10.14 in the special case of averages over concentric balls. More general versions were established by many authors, including Day (1942) and Pitt (1942). The classical methods were pushed to the limit in a notable paper by Tempel’man (1972). Nguyen and Zessin (1979) proved versions of the theorem for finitely additive set functions. The first ergodic theorem for noncommutative transformations was obtained by Zygmund (1951). Sucheston (1983) noted that the statement follows easily from Maker’s (1940) result. In Lemma 10.15, part (i) is due to Rogers and Shephard (1958); part (ii) is elementary.Google Scholar
  4. The ergodic theorem for random matrices was proved by Furstenberg and Kesten (1960), long before the subadditive ergodic theorem became available. The latter result was originally proved by Kingman (1968) under the stronger hypothesis that the array (X m,n) be jointly stationary in m and n. The present extension and shorter proof are due to Liggett (1985).Google Scholar
  5. The ergodic decomposition of invariant measures dates back to Krylov and Bogolioubov (1937), though the basic role of the invariant a-field was not recognized until the work of Farrell (1962) and Varadara-jan (1963). The connection between ergodic decompositions and sufficient statistics is explored in an elegant paper by Dynkin (1978). The traditional approach to the subject is via Choquet theory, as surveyed by Dellacherie and Meyer (1975–87).Google Scholar
  6. The coupling equivalences in Theorem 10.27 (i) were proved by S. Goldstein (1979), after Griffeath (1975) had obtained a related result for Markov chains. The shift coupling part of the same theorem was established by Berbee (1979) and Aldous and Thorisson (1993), and the version for abstract groups was then obtained by Thorisson (1996). The latter author surveyed the whole area in (2000).Google Scholar
  7. Elementary introductions to stationary processes have been given by many authors, beginning with Doob (1953) and Cramer and Lead-better (1967). Loève (1978) contains a more advanced account of probabilistic ergodic theory. A modern and comprehensive survey of the vast area of general ergodic theorems is given by Krengel (1985).Google Scholar

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© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Olav Kallenberg
    • 1
  1. 1.Department of MathematicsAuburn UniversityAuburnUSA

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