π

Relevance to actual stochastic processes and dynamical systems, especially linear and non-linear system identification.

Keywords to look up:

- probability-free ergodicity
- Birkhoff ergodic theorem
- Frobenius-Perron operator
- Quasicompactness, correlation decay
- C&C CLT for Markov chains β Nagaev

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

My main interest is the following 4-stages-of-grief kind of set up.

- Often I can prove that I can learn a thing from my data if it is stationary.
- But I rarely have stationarity, so at least showing the estimator is ergodic might be more useful, which would follow from some appropriate mixing conditions which do not necessarily assume stationarity.
- Except that often these theorems are hard to show, or estimate, or require knowing the parameters in question, and maybe I might suspect that showing some kind of partial identifiability might be more what I need.
- Furthermore, I usually would prefer a finite-sample result instead of some asymptotic guarantee. Sometimes I can get those from learning theory for dependent data.

That last one is TBC.

## Coupling from the past

Dan Piponi does a functional programming explanation of coupling from the past for Markov chains.

## Mixing zoo

A recommended partial overview is BradleyBasic2005. π

### Ξ²-mixing

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### Ο-mixing

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### Sequential Rademacher complexity

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## References

Bradley, Richard C. 2005. βBasic Properties of Strong Mixing Conditions. A Survey and Some Open Questions.β

*Probability Surveys*2: 107β44.
BrΓ©maud, Pierre, and Laurent MassouliΓ©. 2001. βHawkes Branching Point Processes Without Ancestors.β

*Journal of Applied Probability*38 (1): 122β35.
Delft, Anne van, and Michael Eichler. 2016. βLocally Stationary Functional Time Series.β

*arXiv:1602.05125 [Math, Stat]*, February.
Diaconis, Persi, and David Freedman. 1999. βIterated Random Functions.β

*SIAM Review*1 (1): 45β76.
Gray, Robert M. 1987.

*Probability, Random Processes, and Ergodic Properties*. Springer.
Keane, Michael, and Karl Petersen. 2006. βEasy and Nearly Simultaneous Proofs of the Ergodic Theorem and Maximal Ergodic Theorem.β

*IMS Lecture Notes-Monograph Series Dynamics & Stochastics*48.
Kuznetsov, Vitaly, and Mehryar Mohri. 2014. βGeneralization Bounds for Time Series Prediction with Non-Stationary Processes.β In

*Algorithmic Learning Theory*, edited by Peter Auer, Alexander Clark, Thomas Zeugmann, and Sandra Zilles, 260β74. Lecture Notes in Computer Science. Bled, Slovenia: Springer International Publishing.
βββ. 2016. βGeneralization Bounds for Non-Stationary Mixing Processes.β In

*Machine Learning Journal*.
Livan, Giacomo, Jun-ichi Inoue, and Enrico Scalas. 2012. βOn the Non-Stationarity of Financial Time Series: Impact on Optimal Portfolio Selection.β

*Journal of Statistical Mechanics: Theory and Experiment*2012 (07): P07025.
McDonald, Daniel J., Cosma Rohilla Shalizi, and Mark Schervish. 2011. βRisk Bounds for Time Series Without Strong Mixing.β

*arXiv:1106.0730 [Cs, Stat]*, June.
Mohri, Mehryar, and Afshin Rostamizadeh. 2009. βStability Bounds for Stationary Ο-Mixing and Ξ²-Mixing Processes.β

*Journal of Machine Learning Research*4: 1β26.
Morvai, GusztΓ‘v, Sidney Yakowitz, and LΓ‘szlΓ³ GyΓΆrfi. 1996. βNonparametric Inference for Ergodic, Stationary Time Series.β

*The Annals of Statistics*24 (1): 370β79.
Palmer, Richard G. 1982. βBroken Ergodicity.β

*Advances in Physics*31 (6): 669β735.
Propp, James Gary, and David Bruce Wilson. 1996. βExact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics.β In

*Random Structures & Algorithms*, 9:223β52. New York, NY, USA: John Wiley & Sons, Inc.
βββ. 1998. βCoupling from the Past: A Userβs Guide.β In

*Microsurveys in Discrete Probability*, edited by David Aldous and James Gary Propp, 41:181β92. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. Providence, Rhode Island: American Mathematical Society.
Rosenblatt, M. 1984. βAsymptotic Normality, Strong Mixing and Spectral Density Estimates.β

*The Annals of Probability*12 (4): 1167β80.
Ryabko, Daniil, and Boris Ryabko. 2010. βNonparametric Statistical Inference for Ergodic Processes.β

*IEEE Transactions on Information Theory*56 (3): 1430β35.
Shao, Xiaofeng, and Wei Biao Wu. 2007. βAsymptotic Spectral Theory for Nonlinear Time Series.β

*The Annals of Statistics*35 (4): 1773β1801.
Shields, P C. 1998. βThe Interactions Between Ergodic Theory and Information Theory.β

*IEEE Transactions on Information Theory*44 (6): 2079β93.
Steif, Jeffrey E. 1997. βConsistent Estimation of Joint Distributions for Sufficiently Mixing Random Fields.β

*The Annals of Statistics*25 (1): 293β304.
Stein, D L, and C M Newman. 1995. βBroken Ergodicity and the Geometry of Rugged Landscapes.β

*Physical Review E*51 (6): 5228β38.
Thouvenot, Jean-Paul, and Benjamin Weiss. 2012. βLimit Laws for Ergodic Processes.β

*Stochastics and Dynamics*12 (01): 1150012.
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