A Note on the Birkhoff Ergodic Theorem

A Note on the Birkhoff Ergodic Theorem The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behaviour of the time average of a function (having finite p-th moment, $$p\ge 1$$ p ≥ 1 , with respect to the invariant measure) along the trajectories of the process, starting from the invariant measure, is a.s. and in the p-th mean constant and equals to the space average of the function with respect to the invariant measure. The crucial assumption here is that the process starts from the invariant measure, which is not always the case. In this paper, under the assumptions that the underlying process is a Markov process on Polish space, that it admits an invariant probability measure and that its marginal distributions converge to the invariant measure in the $$L^{1}$$ L 1 -Wasserstein metric, we show that the assertion of the Birkhoff ergodic theorem holds in the p-th mean, $$p\ge 1$$ p ≥ 1 , for any bounded Lipschitz function and any initial distribution of the process. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

A Note on the Birkhoff Ergodic Theorem

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0681-9
Publisher site
See Article on Publisher Site

Abstract

The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behaviour of the time average of a function (having finite p-th moment, $$p\ge 1$$ p ≥ 1 , with respect to the invariant measure) along the trajectories of the process, starting from the invariant measure, is a.s. and in the p-th mean constant and equals to the space average of the function with respect to the invariant measure. The crucial assumption here is that the process starts from the invariant measure, which is not always the case. In this paper, under the assumptions that the underlying process is a Markov process on Polish space, that it admits an invariant probability measure and that its marginal distributions converge to the invariant measure in the $$L^{1}$$ L 1 -Wasserstein metric, we show that the assertion of the Birkhoff ergodic theorem holds in the p-th mean, $$p\ge 1$$ p ≥ 1 , for any bounded Lipschitz function and any initial distribution of the process.

Journal

Results in MathematicsSpringer Journals

Published: Apr 26, 2017

References

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