Ergodic Theorems

Krengel, Ulrich

doi:10.1515/9783110844641

Cited by 881 publications

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“…In particular, we see that f belongs to L 1 ( ) for all 2 2 , and therefore from Birkhoff's pointwise ergodic theorem (see [25] p: 9) we have:…”

Section: The Uniform Ergodic Theorem For Dynamical Systemsmentioning

confidence: 94%

“…Then by Birkhoff's pointwise ergodic theorem we have: In addition we have: For proof of all these facts we shall refer the reader to [25] (p.135-136). However, the last one seems to be a weak characterization of (4.13), since there could be too many Banach limits for verification.…”

Section: Corollary 42mentioning

confidence: 96%

“…We consider as before a given ergodic dynamical system (X; B; ; T ) and a parameterized family F = f f j 2 2 g of measurable maps from X into R . We recall that L 2 ( ) is a Hilbert space which is dense in L 1 ( ) , and for which by the general Hilbert space theory we have (see [25] …”

Section: Corollary 42mentioning

confidence: 99%

“…uniformly ergodic (see [25] p.86). Their work generalizes previously established results of Fréchet and Visser, and in particular of Krylov and Bogolioubov whose concept of compact operator is shown to be of vital importance in this direction.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Uniform Ergodic Theorem for Dynamical Systems

Peškir¹,

Weber²

1996

Convergence in Ergodic Theory and Probability

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Necessary and sufficient conditions are given for the uniform convergence over an arbitrary index set in von Neumann's mean and Birkhoff's pointwise ergodic theorem. Three different types of conditions already known from probability theory are investigated. Firstly it is shown that the property of being eventually totally bounded in the mean is necessary and sufficient. This condition involves as a particular case Blum-DeHardt's theorem which offers the best known sufficient condition for the uniform law of large numbers in the independent case. Secondly it is shown that eventual tightness is necessary and sufficient. In this way a link with a weak convergence is obtained. Finally it is shown that the existence of some particular totally bounded pseudo-metrics is necessary and sufficient. The conditions derived are of Lipschitz type, while the method of proof relies upon a result of independent interest, called the uniform ergodic lemma. This result considerably extends Hopf-Yosida-Kakutani's maximal ergodic lemma to a form more suitable for examinations of the uniform convergence under consideration. From this lemma an inequality is also derived which extends the classical (weak) maximal ergodic inequality to the uniform case. In addition, a uniform approximation by means of a dense family of maps satisfying the uniform ergodic theorem in a trivial way is investigated, and a particular result of this type is established. This approach is in the spirit of the classical Hilbert space method for the mean ergodic theorem of von Neumann, and therefore from the ergodic theory point of view it could be seen as the natural one. After this, a simple characterization is obtained for the uniform convergence of moving averages. Finally, a counter-example is constructed for a symmetrization inequality in the stationary ergodic case. This inequality is known to be of vital importance to support the Vapnik-Chervonenkis random entropy approach in the independent case. Further developments in this direction are indicated.

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“…In particular, we see that f belongs to L 1 ( ) for all 2 2 , and therefore from Birkhoff's pointwise ergodic theorem (see [25] p: 9) we have:…”

Section: The Uniform Ergodic Theorem For Dynamical Systemsmentioning

confidence: 94%

Section: Corollary 42mentioning

confidence: 96%

Section: Corollary 42mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Uniform Ergodic Theorem for Dynamical Systems

Peškir¹,

Weber²

1996

Convergence in Ergodic Theory and Probability

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“…The connection of course appears in various disguises in the textbooks [66,97,57]; for expositions dealing more exclusively with large-deviation theory we refer to the books by den Hollander [64], Dembo and Zeitouni [29], and Deuschel and Stroock [30]. For the Pointwise Ergodic Theorem and other facts from ergodic theory we refer to the textbooks by, e.g., Krengel [73] and Petersen [86].…”

Section: Literature Remarksmentioning

confidence: 99%

Reflection Positivity and Phase Transitions in Lattice Spin Models

Biskup¹

2009

Lecture Notes in Mathematics

108

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Law of Large Numbers

Janssen

2005

Encyclopedia of Biostatistics

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It is well known that the sample mean, based on a sequence of independent random variables with common distribution, is a weakly consistent estimator as well as a strongly consistent estimator for the population mean. The first property is the famous weak law of large numbers (Khinchin), the second property is the strong law of large numbers (Kolmogorov). Both laws are key results of modern probability theory. Similar properties hold for more complicated sequences of random variables, for example, for sequences satisfying the stationarity assumption or for martingale sequences. These extensions are of crucial importance for statistical applications in ergodic theory and survival analysis.

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Ergodic Theorems

Cited by 881 publications

References 0 publications

The Uniform Ergodic Theorem for Dynamical Systems

The Uniform Ergodic Theorem for Dynamical Systems

Reflection Positivity and Phase Transitions in Lattice Spin Models

Law of Large Numbers

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