Ergodicity and mixing

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 done! It’s the Pagerank algorithm, more-or-less.
• Quasicompactness, correlation decay
• C&C Nagaev CLT for Markov chains

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 setup.

1. Often I can prove that I can learn a thing from my data if it is stationary.

2. 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.

3. 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.

4. 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.

   But if I can prove nothing it is so bad? Can we prove things for the stochastic process that is the world? Is it bad if not?