Ergodic Theory

A measure on a mathematical space is a way of assigning weights to different parts of the space; volume is a measure on ordinary three-dimensional Euclidean space. Probability distributions are measures, such that the largest measure of any set is 1 (and some other restrictions). Suppose we're interested in a dynamical system --- a transformation that maps a space into itself. The set of points we get from applying the transformation repeatedly to a point is called its trajectory or orbit. Some dynamical systems are measure preserving, meaning that the measure of a set is always the same as the measure of the set of points which map to it. (In symbols, using T for the map and P for the probability measure, for any measureable set A.) Some sets may be invariant: they are the same as their images. An ergodic dynamical system is one in which, with respect to some probability distribution, all invariant sets either have measure 0 or measure 1. (Sometimes non-ergodic systems can be decomposed into a number of components, each of which is separately ergodic.) The dynamics need not be deterministic; in particular, irreducible Markov chains with finite state spaces are ergodic processes, since they have a unique invariant distribution over the states. (In the Markov chain case, each of the ergodic components corresponds to an irreducible sub-space.)

Ergodicity is important because of the following theorem (due to von Neumann, and then improved substantially by Birkhoff, in the 1930s). If we take any well-behaved (integrable) function of our space, pick a point in the space at random (according to the ergodic distribution) and calculate the average of the function along the point's orbit, the time-average. Then, with probability 1, in the limit as the time goes to infinity, (1) the time-average converges to a limit and (2) that limit is equal to the weighted average of the value of the function at all points in the space (with the weights given by the same distribution), the space-average. The orbit of almost any point you please will in some sense look like the whole of the state space.

(Symbolically, write x for a point in the state space, f for the function we're averaging, and T and P for the map and the probability measure as before. The space-average, . The time-average starting from x, . The ergodic theorem asserts that if f is integrable and T is ergodic with respect to P, then exists, and . --- A similar result holds for continuous-time dynamical systems, where we replace the summation in the time average with an integral.)

This is an extremely important property for statistical mechanics. In fact, the founder of statistical mechanics, Ludwig Boltzmann, coined "ergodic" as the name for a stronger but related property: starting from a random point in state space, orbits will typically pass through every point in state space. It is easy to show (with set theory) that this isn't doable, so people appealled to a weaker property which was for a time known as "quasi-ergodicity": a typical trajectory will pass arbitrarily close to every point in phase space. Finally it became clear that only the modern ergodic property is needed. To see the relation, consider the function, call it I, which is 1 on a certain set, call it A, and 0 elsewhere. The time-average of I is the fraction of time that the orbit spends in A. The space-average of I is the probability that a randomly picked point is in A. Since the two averages are almost always equal, almost all trajectories end up covering the state space in the same way.

One way of thinking about the classical ergodic theorem is that it's a version of the law of large numbers --- it tells us that a sufficiently large sample (i.e., an average over a long time) is representative of the whole population (i.e., the space average). One thing I'd like to know more about than I do is ergodic equivalents of the central limit theorem, which say how big the sampling fluctuations are, and how they're distributed. The other thing I want to know about is the rate of convergence in the ergodic theorem --- how long must I wait before my time average is within a certain margin of probable error of the state average. Here I do know a bit more of the relevant literature, from large deviations theory.

Again in symbols: Let's write for the time-average of f, starting from x, taken over n steps. Then a central-limit theorem result would say that (for example) converges in distribution to a Gaussian with mean zero and variance one, where is the (space-averaged) variance of f and is some positive, increasing function of n. This would be weak convergence of the time averages to the space averages, and would give the rate. (In the usual IID case, .) Somewhat stronger would be a convegence in probability result, giving us a function such that if . Proving many of these results requires stronger assumptions than proving ergodicity does --- for instance, Markov properties, or mixing properties (which I should explain here, but won't).

These issues are part of a more general question about how to do statistical inference for stochastic processes, a.k.a. time-series analysis. I am especially interested in statistical learning theory in this setting.

Another thing I need to understand, but don't have time to explain here, are Pinsker sigma-algebras.

Peter Billingsley, Ergodic Theory and Information
Robert M. Gray, Probability, Random Processes, and Ergodic Properties [Full-text online]
A. I. Khinchin, Mathematical Foundations of Statistical Mechanics [Proves the von Neumann-Birkhoff ergodic theorem in detail]
Mark Kac, Probability and Related Topics in Physical Science
Andrzej Lasota and Michael C. Mackey, Chaos, Fractals and Noise: Stochastic Aspects of Dynamics
Norbert Wiener, Cybernetics: Or Control and Communication in the Animal and the Machine

J.-R. Chazottes and R. Leplaideur, "Birkhoff averages of Poincare cycles for Axiom-A diffeomorphisms," math.DS/0312291
Jérôme Dedecker, Paul Doukhan, Gabriel Lang, José Rafael León R., Sana Louhichi and Clémentine Prieur, Weak Dependence: With Examples and Applications [Mini-review]
Paul Doukhan, Mixing: Properties and Examples
E. B. Dynkin, "Sufficient statistics and extreme points", Annals of Probability 6 (1978): 705--730 ["The connection between ergodic decompositions and sufficient statistics is explored in an elegant paper by DYNKIN" --- Kallenberg, Foundations of Modern Probability, p. 577]
Jean-Pierre Eckmann and David Ruelle, "Ergodic Theory of Chaos and Strange Attractors," Reviews of Modern Physics 57 (1985): 617--656
Roberto Fernández and Grégory Maillard, "Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties", Journal of Statistical Physics 118 (2005): 555--588
Stefano Galatolo, Mathieu Hoyrup, and Cristóbal Rojas, "Effective symbolic dynamics, random points, statistical behavior, complexity and entropy", arxiv:0801.0209 [All, not almost all, Martin-Löf points are statistically typical.]
Weihong Huang, "On the long-run average growth rate of chaotic systems", Chaos 14 (2004): 38--47 [An amusing demonstration that positive-valued ergodic processes will seem to always have a positive long-run growth rate, even though they're stationary!]
Michael Keane and Karl Petersen, "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", math.DS/0608251 [This is a lovely little four-page paper, and the simplest proof by far that I've seen, but they do rely rather heavily on the reader being familiar with facts about time averages, invariant functions, etc. Still, I should definitely teach this in my class.]
Aryeh (Leo) Kontorovich, "Metric and Mixing Sufficient Conditions for Concentration of Measure", math.PR/0610427 [See weblog comments]
Aryeh Kontorovich and Anthony Brockwell, "A Strong Law of Large Numbers for Strongly Mixing Processes", arxiv:0807.4665
Michael C. Mackey, Time's Arrow: The Origins of Thermodynamic Behavior
Florence Merlevède, Magda Peligrad, Sergey Utev, "Recent advances in invariance principles for stationary sequences", math.PR/0601315 = Probability Surveys 3 (2006): 1--36 [Can I just say how much I hate calling the functional central limit theorem "the invariance principle"?]
Mehryar Mohri and Afshin Rostamizadeh, "Stability Bound for Stationary Phi-mixing and Beta-mixing Processes", arxiv:0811.1629 ["Stability" is the property of a learning algorithm that changing a single observation in the training set leads to only small changes in predictions on the test set. This paper shows that stable learning algorithms continue to perform well with dependent data, provided the data are either phi mixing or beta mixing. The results for phi-mixing depend on Leo's work.]
Andrew Nobel and Amir Dembo, "A Note on Uniform Laws of Averages for Dependent Processes", Statistics and Probability Letters 17 (1993): 169--172 [An extremely easy way to extend uniform laws of large numbers to uniform ergodic theorems for mixing processes. Actually I suspect that mixing is only necessary to get an explicit rate; I should re-read it. PDF preprint via Dr. Nobel.]
Donald Ornstein and Benjamin Weiss, "How Sampling Reveals a Process", Annals of Probability 18 (1990): 905--930 [Open access. Some comments under Universal Prediction.]
Murray Rosenblatt, "A Central Limit Theorem and a Strong Mixing Condition", Proceedings of the National Academy of Sciences (USA) 42 (1956): 43--47 [The root from which much subsequent ergodic theory has sprung. PDF reprint]
Daniil Ryabko and Boris Ryabko, "Testing Statistical Hypotheses About Ergodic Processes", arxiv:0804.0510
Nobusumi Sagara, "Nonparametric maximum-likelihood estimation of probability measures: existence and consistency", Journal of Statistical Planning and Inference 133 (2005): 249--271 ["This paper formulates the nonparametric maximum-likelihood estimation of probability measures and generalizes the consistency result on the maximum-likelihood estimator (MLE). We drop the independent assumption on the underlying stochastic process and replace it with the assumption that the stochastic process is stationary and ergodic. The present proof employs Birkhoff's ergodic theorem and the martingale convergence theorem. The main result is applied to the parametric and nonparametric maximum-likelihood estimation of density functions." Very cool.]
Paul C. Shields, The Ergodic Theory of Discrete Sample Paths [Well-written modern text, extremely strong on connections to information theory and coding. I haven't gotten through the last chapter, however. Shield's page on the book.]
Leslie Sklar, Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics [Good discussion of ergodic results in several places]
Marta Tyran-Kaminska, "An Invariance Principle for Maps with Polynomial Decay of Correlations", math.DS/0408185 = Communications in Mathematical Physics 260 (2005): 1--15 ["We give a general method of deriving statistical limit theorems, such as the central limit theorem and its functional version, in the setting of ergodic measure preserving transformations. This method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations."]
Mathukumalli Vidyasagar, A Theory of Learning and Generalization: With Applications to Neural Networks and Control Systems [Has a very nice discussion of when the uniform laws of large numbers of statistical learning theory transfer from the usual IID setting to dependent processes, becoming uniform ergodic theorems. (Sufficient conditions include things like beta-mixing, but necessary and sufficient conditions seem to still be unknown.) Mini-review]
Benjamin Weiss, Single Orbit Dynamics
Wei Biao Wu, "Nonlinear system theory: Another look at dependence", Proceedings of the National Academy of Sciences 102 (2005): 14150--14154 ["we introduce [new] dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms." Proofs rely heavily on results from Wu's other papers, which I have yet to read.]

chs. 5 and 22--27 of Almost None of the Theory of Stochastic Processes

Jon Aaronson, An Introduction to Infinite Ergodic Theory [Blurb]
Jon Aaronson and Tom Meyerovitch, "Absolutely continuous, invariant measures for dissipative, ergodic transformations", math.DS/0509093 ["We show that a dissipative, ergodic measure preserving transformation of a sigma-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these."]
Jose M. Amigo, Matthew B. Kennel and Ljupco Kocarev, "The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems", nlin.CD/0503044
Vitor Araujo, "Semicontinuity of entropy, existence of equilibrium states and of physical measures", math.DS/0410099
L. Arnold, Random Dynamical Systems
V. I. Arnol'd and A. Avez, Ergodic Problems of Classical Mechanics
Jeremy Avigad, Philipp Gerhardy and Henry Towsner, "Local stability of ergodic averages", arxiv:0706.1512 [Computing bounds on the rate of convergence in the ergodic theorems; sound scool. Thanks to Gustavo Lacerda for the pointer.]
Massimiliano Badino, "The Foundational Role of Ergodic Theory", phil-sci/2277
Dominique Bakry, Patrick Cattiaux, Arnaud Guillin, "Rate of Convergence for ergodic continuous Markov processes : Lyapunov versus Poincare", math.PR/0703355
Viviane Baladi, Positive Transfer Operators and Decay of Correlations
Joseph Berkovitz, Roman Frigg and Fred Kronz, "The Ergodic Hierarchy, Randomness and Hamiltonian Chaos", phil-sci/2927
A. A. Borovkov, Ergodicity and Stability of Stochastic Processes
J.-R. Chazottes and P. Collet, "Almost-sure central limit theorems and the Erdös-Rényi law for expanding maps of the interval", Ergodic Theory and Dynamical Systems 25 (2005): 419--41
J.-R. Chazottes, P. Collet and B. Schmitt, "Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems", math.DS/0412166 = Nonlinearity 18 (2005): 2323--2340
J.-R. Chazottes, P. Collet and B. Schmitt, "Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems", math.DS/0412167 = Nonlinearity 18 (2005): 2341--2364
J.-R. Chazottes and G. Gouezel, "On almost-sure versions of classical limit theorems for dynamical systems", math.DS/0601388 [arguing in support of the idea that "whenever we can prove a limit theorem in the classical sense for a dynamical system, we can prove a suitable almost-sure version based on an empirical measure with log-average".]
J.-R. Chazottes and F. Redig, "Testing the irreversibility of a Gibbsian process via hitting and return times", math-ph/0503071 = Nonlinearity 18 (2005): 2477--2489
Mu-Fa Chen
- Eigenvalues, Inequalities, and Ergodic Theory [Blurb]
- "Ergodic convergence rates of Markov processes--eigenvalues, inequalities and ergodic theory", math.PR/0304367
Xia Chen, Limit Theorems for Functionals of Ergodic Markov Chains with General State Space
Geon Ho Choe, Computational Ergodic Theory
Yves Coudene, "On invariant distributions and mixing", Ergodic Theory and Dynamical Systems 27 (2007): 109--112 ["any probability preserving transformation of a metric space is mixing as soon as there are no non-constant $L^2$-functions which are invariant under both the stable and unstable distributions"]
Thierry De La Rue, "An introduction to joinings in ergodic theory", math.DS/0507429 = Discrete and Continuous Dynamical Systems 15 (2006): 121--142
G. B. DiMasi and L. Stettner, "Ergodicity of hidden Markov models", Mathematics of Control, Signals, and Systems 17 (2005): 269--296
Ian Domowitz and Mahmoud El-Gamal, "A Consistent Nonparametric Test of Ergodicity for Time Series with Applications", Journal of Econometrics 102 (2001): 365--398 [SSRN. Having read about 2/3 of this, I completely fail to see how they actually overcome the problem that any one sample path is always confined to a single ergodic component.]
Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Russell Martin, "Markov chain comparison", math.PR/0410331 [i.e., comparison theorems for mixing times]
Jean-Pierre Eckmann and Itamar Procaccia, "Invariant Measures in Generic Dynamical Systems", chao-dyn/9708021 [Abstract: "Irreversible thermodynamics of simple fluids have been connected recently to the theory of dynamical systems and some interesting assumptions have been made about the nature of the associated invariant measures. We show that the tests of the validity of these assumptions are insufficient by exhibiting observables that are incorrectly sampled with the proposed invariant measures. Only observables belonging to the 'high temperature phase' of the thermodynamic formalism are insensitive to the sampling methods. We outline methods that are free of these deficiencies." I read this when it came out, but I don't think I understood it then.]
Bernhold Fiedler (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems
Nikos Frantzikinakis, Randall McCutcheon, "Ergodic Theoy: Recurrence", arxiv:0705.0033
Gary Froyland, "Statistical optimal almost-invariant sets", Physica D 200 (2005): 205--219 [Partitioning state space into nearly separated components.]
Stefano Galatolo, Mathieu Hoyrup, Cristobal Rojas, "Dynamics and abstract computability: computing invariant measures", arxiv:0903.2385
Gan Shixin, "Almost sure convergence for -mixing random variable sequences", Statistics and Probability Letters 67 (2004): 289--298
E. Glasner and B. Weiss, "On the interplay between measurable and topological dynamics", math.DS/0408328
Beniamin Goldys and Bohdan Maslowski, "Uniform exponential ergodicity of stochastic dissipative systems," math.PR/0111143
M. Hairer, "Ergodic properties of a class of non-Markovian processes", arxiv:0708.3338
P. Halmos, Ergodic Theory
Nicolai T. A. Haydn, "The Central Limit Theorem for uniformly strong mixing measures", arxiv:0903.1325
Nicolai Haydn, Y. Lacroix and Sandro Vaienti, "Hitting and return times in ergodic dynamical systems", math.DS/0410384 = Annals of Probability 33 (2005): 2043--2050
Nicolai Haydn and Sandro Vaienti, "Fluctuations of the Metric Entropy for Mixing Measures", Stochastics and Dynamics 4 (2004): 595--627
Bernard Host, "Convergence of multiple ergodic averages", math.DS/0606362 ["We study the mean convergence of multiple ergodic averages, that is, averages of a product of functions taken at different times."]
Gerhard Keller, Equilibrium States in Ergodic Theory
Gerhard Keller and Carlangelo Liverani, "Uniqueness of the SRB Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension", Communications in Mathematical Physics 262 (2006): 33--50
U. Kregnel, Ergodic Theorems
Anna Kuczmaszewska, "The strong law of large numbers for dependent random variables", Statistics and Probability Letters 73 (2005): 305--314
B. Kümmerer and H. Maassen, "A pathwise ergodic theorem for quantum trajectories", Journal of Physics A 37 (2004): 11889--11896 [journal link]
Lin Zhengyan and Lu Chuanrong, Limit Theory for Mixing Dependent Random Variables
Pei-Dong Liu and Min Qian, Smooth Ergodic Theory of Random Dynamical Systems
Dasha Loukianova, Oleg Loukianov, Eva Loecherbach, "Polynomial bounds in the Ergodic Theorem for positive recurrent one-dimensional diffusions and integrability of hitting times", arxiv:0903.2405 [non-asymptotic deviation bounds from bounds on moments of recurrence times]
Stefano Luzzatto
- "Mixing and decay of correlations in non-uniformly expanding maps: a survey of recent results," math.DS/0301319
- "Stochastic-like behaviour in nonuniformly expanding maps", math.DS/0409085
Stefano Luzzatto, Ian Melbourne and Frederic Paccaut, "The Lorenz Attractor is Mixing", Communications in Mathematical Physics 260 (2005): 393--401
Vincent Lynch, "Decay of correlations for non-Holder observables", math.DS/0401432
Michael C. Mackey and Marta Tyran-Kaminska
- "Deterministic Brownian Motion: The Effects of Perturbing a Dynamical System by a Chaotic Semi-Dynamical System", cond-mat/0408330
- "Effects of Noise on Entropy Evolution", cond-mat/0501092
- "Central Limit Theorems for Non-Invertible Measure Preserving Maps", math.PR/0608637 ["a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator"]
C. Maes, F. Redig and E. Saada, "The Infinite Volume Limit of Dissipative Abelian Sandpiles", Communications in Mathematical Physics 244 (2004): 395--417
Katalin Marton and Paul C. Shields, "How many future measures can there be?", Ergodic Theory and Dynamical Systems 22 (2002): 257--280
Jonathan C. Mattingly, "On Recent Progress for the Stochastic Navier Stokes Equations", math.PR/0409194 ["We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations."]
Ian Melbourne and Matthew Nicol
- "Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems", Communications in Mathematical Physics 260 (2005): 131--146 = math.DS/0503693 ["We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. ... Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences."]
- "A Vector-Valued Almost Sure Invariance Principle for Hyperbolic Dynamical Systems", math.DS/0606535
Mehryar Mohri, Afshin Rostamizadeh, "Stability Bound for Stationary Phi-mixing and Beta-mixing Processes", arxiv:0811.1629
D. S. Ornstein and B. Weiss, "Statistical Properties of Chaotic Systems," Bulletin of the American Mathematical Society 24 (1991): 11--116
Goran Peskir, From Uniform Laws of Large Numbers to Uniform Ergodic Theorems
Karl E. Petersen, Ergodic Theory
Mark Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds
Mark Pollicott and Michiko Yuri, Dynamical Systems and Ergodic Theory [PDF files for the individual chapters are available here, but many of them don't display properly, at least on my laptop...]
Charles Pugh and Michael Shub, with an appendix by Alexander Starkov, "Stable Ergodicity", Bulletin of the American Mathematical Society (new series) 41 (2003): 1--41 [Link]
Maxim Raginsky, "Joint universal lossy coding and identification of stationary mixing sources with general alphabets", arxiv:0901.1904
Gennady Samorodnitsky, "Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes", Annals of Probability 32 (2004): 1438--1468 = math.PR/0410149
C. E. Silva, Invitation to Ergodic Theory [blurb]
Ya. Sinai, Topics in Ergodic Theory
Rob Sturman, Julio M. Ottino, and Stephen Wiggins, The Mathematical Foundations of Mixing: The Linked Twist Map as a Paradigm in Applications: Micro to Macro, Fluids to Solids [Blurb]
Andre Toom
- "Law of Large Numbers for Non-Local Functions of Probabilistic Cellular Automata", Journal of Statistical Physics 133 (2008): 883--897
- "Every Continuous Operator Has an Invariant Measure", Journal of Statistical Physics 129 (2007): 555--566
A. Vershik, "Towards the definition of metric hyperbolicity", math.DS/0508514
Peter Walters, An Introduction to Ergodic Theory
Wei Wang, Jianhua Sun and Jinqiao Duan, "Ergodic Dynamics of the Stochastic Swift-Hohenberg System", math.DS/0408322
Wei Biao Wu, Xiaofeng Shao, "Invariance principles for fractionally integrated nonlinear processes", math.PR/0608223
Wei Biao Wu and Michael Woodroofe, "Martingale Approximations for Sums of Stationary Processes", Annals of Probability 32 (2004): 1674--1690 = math.PR/0410160
Ivan Werner, "Ergodic theorem for contractive Markov systems", Nonlinearity 17 (2303--2313) [PS preprint]
Guangyu Yang, Yu Miao, "An invariance principle for the law of the iterated logarithm for additive functionals of Markov chains", math.PR/0609593
Radu Zaharopol, Invariant Probabilities of Markov-Feller Operators and Their Supports
Steve Zelditch, "Quantum ergodicity and mixing", quant-ph/0503026 ["an expository article for the Encyclopedia of Mathematical Physics"]
L. Zsido, "Weaking mixing properties of vector sequences", math.FA/0506554

Updated 29 October 2007; thanks to "tushar" for pointing out an embarrassing think-o in the first paragraph.

Previous versions: 15 Nov 2005 16:18; first version written c. 1997 (?)

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