Abstract
The basic ergodic theorems are proved showing that asymptotic mean stationarity is a sufficient as well as necessary condition for a random process or dynamical system to have convergent sample averages for all bounded measurements. Conditions are found for the convergence of unbounded sample averages in both the almost everywhere and mean sense. The proof follows the coding-flavored proofs developed in the ergodic theory literature in the 1970s rather than the classical route of the maximal ergodic theorem.
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© 2009 Springer-Verlag US
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Gray, R.M. (2009). Ergodic Theorems. In: Probability, Random Processes, and Ergodic Properties. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1090-5_8
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DOI: https://doi.org/10.1007/978-1-4419-1090-5_8
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Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-1089-9
Online ISBN: 978-1-4419-1090-5
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