Abstract
In the context of stochastic processes, ergodic theory relates the long-run “time-averages” such as the sample mean of an evolving strictly stationary process X 0, X 1, … to a “phase-average” computed as an expected value with respect to a probability distribution on the state space. This is the perspective developed in this chapter.
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Notes
- 1.
This result, which was motivated by considerations of the relationship between “time-averages” and “phase-averages” in statistical physics and dynamical systems, is due to Birkhoff (1931).
- 2.
See BCPT, p. 87.
- 3.
The derivation presented here follows Garcia (1965).
- 4.
See BCPT, p. 17 for this L 1-convergence criteria.
- 5.
Bhattacharya and Waymire (2021), p. 162.
- 6.
An extension of de Finetti’s theorem to exchangeable Markov processes was initiated in Diaconis and Freedman (1980) that is worthy of mention here. Especially see James et al. (2008) for inspiring connections to transient random walk.
- 7.
See BCPT p.4.
- 8.
See BCPT p.168.
- 9.
See Spitzer (1964), p. 38., where the result is attributed to Kesten, H., F. Spitzer, and W. Whitman. This result had been obtained for the k-dimensional simple symmetric random walk in an earlier paper by Dvoretzky and Erdos (1951).
- 10.
See BCPT, p.237.
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Bhattacharya, R., Waymire, E. (2022). Birkhoff’s Ergodic Theorem. 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_4
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