Relevance to actual stochastic processes and dynamical systems, especially linear. and nonlinear system identification, and long-memory systems.
Keywords to look up:
- probability-free ergodicity
- Birkhoff ergodic theorem
- Frobenius-Perron operator
- Quasicompactness, correlation decay
- C&C CLT for Markov chains - Nagaev
- Coupling from the past
Not much material here, but please see learning theory for dependent data for some interesting categorisations of mixing and transcendence of miscellaneous mixing conditions for statistical estimators.
Coupling from the past
Dan Piponi does a functional programming explanation of coupling from the past for markov chains.
Sequential Rademacher complexity
- Rose84: (1984) Asymptotic Normality, Strong Mixing and Spectral Density Estimates. The Annals of Probability, 12(4), 1167–1180. DOI
- ShWu07: (2007) Asymptotic spectral theory for nonlinear time series. The Annals of Statistics, 35(4), 1773–1801. DOI
- Palm82: (1982) Broken ergodicity. Advances in Physics, 31(6), 669–735. DOI
- StNe95: (1995) Broken ergodicity and the geometry of rugged landscapes. Physical Review E, 51(6), 5228–5238. DOI
- Stei97: (1997) Consistent estimation of joint distributions for sufficiently mixing random fields. The Annals of Statistics, 25(1), 293–304.
- PrWi98: (1998) Coupling from the past: a user’s guide. In Microsurveys in Discrete Probability (Vol. 41, pp. 181–192). Providence, Rhode Island: American Mathematical Society DOI
- KePe06: (2006) Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem. IMS Lecture Notes-Monograph Series Dynamics & Stochastics, 48. DOI
- PrWi96: (1996) Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics. In Random Structures & Algorithms (Vol. 9, pp. 223–252). New York, NY, USA: John Wiley & Sons, Inc. DOI
- BrMa01: (2001) Hawkes branching point processes without ancestors. Journal of Applied Probability, 38(1), 122–135. DOI
- DiFr99: (1999) Iterated Random Functions. SIAM Review, 41(1), 45–76. DOI
- ThWe12: (2012) Limit Laws for Ergodic Processes. Stochastics and Dynamics, 12(01), 1150012. DOI
- MoYG96: (1996) Nonparametric inference for ergodic, stationary time series. The Annals of Statistics, 24(1), 370–379. DOI
- Lair78: (1978) Nonparametric Maximum Likelihood Estimation of a Mixing Distribution. Journal of the American Statistical Association, 73(364), 805–811. DOI
- RyRy10: (2010) Nonparametric Statistical Inference for Ergodic Processes. IEEE Transactions on Information Theory, 56(3), 1430–1435. DOI
- Whit54: (1954) On stationary processes in the plane. Biometrika, 41(3/4), 434–449.
- LiIS12: (2012) On the non-stationarity of financial time series: impact on optimal portfolio selection. Journal of Statistical Mechanics: Theory and Experiment, 2012(07), P07025. DOI
- Gray09: (2009) Probability, random processes, and ergodic properties. Springer Verlag
- McSS11: (2011) Risk bounds for time series without strong mixing. ArXiv:1106.0730 [Cs, Stat].
- GöKü96: (1996) Second-order correctness of the blockwise bootstrap for stationary observations. The Annals of Statistics, 24(5), 1914–1933. DOI
- AlBo05: (2005) Stationary solutions for integer-valued autoregressive processes. International Journal of Mathematics and Mathematical Sciences, 2005(1), 1–18. DOI
- Shie98: (1998) The interactions between ergodic theory and information theory. IEEE Transactions on Information Theory, 44(6), 2079–2093. DOI