## Mixing and Weak Dependence of Stochastic Processes

*08 Dec 2018 11:19*

Yet Another Inadequate Placeholder. I am mostly using this to collect references on different concepts of asymptotic independence in stochastic processes, and not so much the consequences which follow from these.

See also: Deviation Inequalities; Ergodic Theory; Independence Tests and Dependence Measures; Statistical Learning with Dependent Data; Stochastic Processes;

- Recommended, big picture:
- Richard C. Bradley, "Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions", Probability Surveys
**2**(2005): 107--144, arxiv:math/0511078 - 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
- Andrzej Lasota and Michael C. Mackey, Chaos, Fractals and Noise: Stochastic Aspects of Dynamics [For the very careful treatment of the hierarchy of mixing properties, at least as viewed from ergodic theory and dynamics]

- Recommended, close-ups:
- J.-R. Chazottes, P. Collet, C. Kuelske and F. Redig, "Deviation inequalities via coupling for stochastic processes and random fields", math.PR/0503483 [Very cool]
- V. Maume-Deschamps, "Concentration Inequalities and Estimation of Conditional Probabilities" [PDF]
- 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] - 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."] - 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.] - Bin Yu
- "Density Estimation in the $L^{\infty}$ Norm for Dependent Data with Applications to the Gibbs Sampler", Annals of Statistics
**21**(1993): 711--735 - "Rates of Convergence for Empirical Processes of Stationary
Mixing Sequences", Annals of Probability
**22**(1994): 94--116

- "Density Estimation in the $L^{\infty}$ Norm for Dependent Data with Applications to the Gibbs Sampler", Annals of Statistics

- Modesty forbids me to recommend:
- Daniel J. McDonald, CRS and Mark Schervish
- "Estimating beta-mixing coefficients", AISTATS 2011
- "Estimating Beta-Mixing Coefficients via Histograms",
Electronic Journal of
Statistics
**9**(2015): 2855--2883, arxiv:1109.5998

- CRS and Aryeh Kontorovich, "Predictive PAC Learning and Process Decompositions", NIPS 2013, pp. 1619--1627, arxiv:1309.4859

- To read:
- Roberto Artuso, Cesar Manchein, "Instability statistics and mixing rates", arxiv:0906.0791
- Patrice Bertail, Paul Doukhan and Philippe Soulier (eds.), Dependence in Probability and Statistics ["recent developments in ... probability and statistics for dependent data... from Markov chain theory and weak dependence with an emphasis on ... dynamical systems, to strong dependence in times series and random fields. ... section on statistical estimation problems and specific applications". Full blurb, contents]
- Alberto Bressan, "A Lemma and a Conjecture on the Cost of Rearrangements", Rendiconti del Seminario Matematico della Università di Padova
**110**(2003): 97--102 - 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"] - Łukasz Dębowski, "Mixing, Ergodic, and Nonergodic Processes with Rapidly Growing Information between Blocks", arxiv:1103.3952
- Herold Dehling and Olivier Durieu, "Empirical Processes of Multidimensional Systems with Multiple Mixing Properties", arxiv:1004.1088
- Victor H. de la Pena, Rustam Ibragimov, and Shaturgun Sharakhmetov, "Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series", math.ST/0611166
- Nicolas Fournier, Arnaud Guillin, "On the rate of convergence in Wasserstein distance of the empirical measure", arxiv:1312.2128
- Piotr Fryzlewicz, Suhasini Subba Rao, "Mixing properties of ARCH and time-varying ARCH processes", Bernoulli
**17**(2011): 320--346, arxiv:1102.2053 - A. Guionnet and B. Zegarlinski, Lectures on Logarithmic Sobolev Inequalities [120 pp. PDF]
- H. Hang, I. Steinwart, "A Bernstein-type Inequality for Some Mixing Processes and Dynamical Systems with an Application to Learning", arxiv:1501.03059
- Leonid (Aryeh) Kontorovich, "Constructing processes with prescribed mixing coefficients", arxiv:0711.0986
- S. N. Lahiri, "Edgeworth expansions for studentized statistics under weak dependence", Annals of Statistics
**38**(2010): 388--434 - Remi Peyre, "Tensorizing maximal correlations", arxiv:1004.1602
- Patrick Rebeschini, Ramon van Handel, "Comparison Theorems for Gibbs Measures", Journal of Statistical Physics
**157**(2014): 234--281, arxiv:1308.4117 - Emmanuel Rio, Asymptotic Theory of Weakly Dependent Random Processes