Notebooks

## Mixing and Weak Dependence of Stochastic Processes

24 Jan 2018 15:35

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.

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.]
• 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"]
• 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