# Ergodic theory / mixing

TBD.

Relevance to actual stochastic processes and dynamical systems, especially linear. and nonlinear system identification, and long-memory systems.

Keywords to look up:

• Birkhoff ergodic theorem.
• Frobenius-Perron operator.
• Quasicompactness, correlation decay.
• C&C CLT for Markov chains - Nagaev.

Not much material here, but please see learning theory for dependent data for some interesting categorisations of mixing and transcendence of stupid mixing conditions.

## Refs

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Gallo, S., & Leonardi, F. G.(2014) Nonparametric statistical inference for the context tree of a stationary ergodic process. arXiv:1411.7650 [Math, Stat].
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Götze, F., & Künsch, H. R.(1996) Second-order correctness of the blockwise bootstrap for stationary observations. The Annals of Statistics, 24(5), 1914–1933. DOI.
Gray09
Gray, R. M.(2009) Probability, random processes, and ergodic properties. . Springer Verlag
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Ishioka, S., & Fuchikami, N. (2001) Thermodynamics of computing: Entropy of nonergodic systems. Chaos, 11(3), 734–746. DOI.
KePe06
Keane, M., & Petersen, K. (2006) Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem. IMS Lecture Notes-Monograph Series Dynamics & Stochastics, 48. DOI.
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Laird, N. (1978) Nonparametric Maximum Likelihood Estimation of a Mixing Distribution. Journal of the American Statistical Association, 73(364), 805–811. DOI.
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McSS11
McDonald, D. J., Shalizi, C. R., & Schervish, M. (2011) Risk bounds for time series without strong mixing. arXiv:1106.0730 [Cs, Stat].
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Morvai, G., Yakowitz, S., & Györfi, L. (1996) Nonparametric inference for ergodic, stationary time series. The Annals of Statistics, 24(1), 370–379. DOI.
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Palmer, R. G.(1982) Broken ergodicity. Advances in Physics, 31(6), 669–735. DOI.
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Rosenblatt, M. (1984) Asymptotic Normality, Strong Mixing and Spectral Density Estimates. The Annals of Probability, 12(4), 1167–1180. DOI.
RyRy10
Ryabko, D., & Ryabko, B. (2010) Nonparametric Statistical Inference for Ergodic Processes. IEEE Transactions on Information Theory, 56(3), 1430–1435. DOI.
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Shao, X., & Wu, W. B.(2007) Asymptotic spectral theory for nonlinear time series. The Annals of Statistics, 35(4), 1773–1801. DOI.
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Shields, P. C.(1998) The interactions between ergodic theory and information theory. Information Theory, IEEE Transactions on, 44(6), 2079–2093. DOI.
Stei97
Steif, J. E.(1997) Consistent estimation of joint distributions for sufficiently mixing random fields. The Annals of Statistics, 25(1), 293–304.
StNe95
Stein, D. L., & Newman, C. M.(1995) Broken ergodicity and the geometry of rugged landscapes. Physical Review E, 51(6), 5228–5238. DOI.
ThWe12
THOUVENOT, J.-P., & Weiss, B. (2012) Limit Laws for Ergodic Processes. Stochastics and Dynamics, 12(1), 1150012. DOI.
Whit54
Whittle, P. (1954) On stationary processes in the plane. Biometrika, 41(3/4), 434–449.