Abstract
For discrete parameter Markov processes on a general state space, Birkhoff’s ergodic theorem provides a natural approach to the existence of invariant probabilities and the corresponding strong law of large numbers in some generality. In addition, it is shown that the notion of an irreducible positive recurrent Markov chain on a countable state space is equivalent to being irreducible ergodic stationary Markov chain having a unique invariant initial distribution.
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References
Bhattacharya RN (1982) On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 60(2):185–201
Gordin MI, Lifsic D (1978) The central limit theorem for stationary, ergodic Markov processes. Doklady 19:392–394
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Bhattacharya, R., Waymire, E. (2022). Stationary Ergodic Markov Processes: SLLN & FCLT. In: Stationary Processes and Discrete Parameter Markov Processes. Graduate Texts in Mathematics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-031-00943-3_16
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DOI: https://doi.org/10.1007/978-3-031-00943-3_16
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