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The term "strong mixing conditions" plural can reasonably be thought of as referring to all conditions that are at least as strong as i. Each of these four sequences of dependence coefficients is trivially nonincreasing.

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As a consequence of these inequalities and some well known examples, one has the following "hierarchy" of the five strong mixing conditions here:. For all of these mixing conditions, the "mixing rates" can be essentially arbitrary, and in particular, arbitrarily slow. That general principle was established by Kesten and O'Brien in with several classes of examples.

Weak Dependence: With Examples and Applications

For further details, see e. The various strong mixing conditions above have been used extensively in statistical inference for weakly dependent data. See e. Ibragimov's conjecture and related material. In the s, I.

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Technically, this conjecture remains unsolved. For this and related material, see e. There is a vast literature on central limit theory for random fields satisfying various strong mixing conditions. In the formulation of mixing conditions for random fields and also "interlaced" mixing conditions for random sequences some caution is needed; see e.

Connections with specific types of models.

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Now let us return briefly to a theme from the beginning of this write-up: the connection between strong mixing conditions and specific structures. Markov chains. For this and other information on strong mixing conditions for Markov chains, see e. Stationary Gaussian sequences. Dynamical systems. Many dynamical systems have strong mixing properties.

For certain stationary finite-state stochastic processes built on piecewise expanding mappings of the unit interval onto itself, the absolute regularity condition holds with at least exponentially fast mixing rate. For more detains on the mixing properties of these and other dynamical systems, see e.

Denker [De]. Linear and related processes. There is a large literature on strong mixing properties of strictly stationary linear processes including strictly stationary ARMA processes and also "non-causal" linear processes and linear random fields and also of some other related processes such as bilinear, ARCH, or GARCH models. For details on strong mixing properties of these and other related processes, see e.

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Doukhan [Do, Chapter 2]. For more on this example, see e. Further related developments. There is a substantial development of central limit theory for strictly stationary sequences under weak dependence assumptions explicitly involving characteristic functions in connection with "block sums"; much of that theory is codified in [Ja]. In recent years, there has been a considerable development of central limit theory under "projective" criteria related to martingale theory motivated by Gordin's martingale-approximation technique see [HH] ; for details, see e.

There are far too many other types of weak dependence conditions, of the general spirit of strong mixing conditions but less restrictive, to describe here; for more details, see e. Log in. Namespaces Page Discussion.

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My profile My library Metrics Alerts. Sign in. Get my own profile Cited by View all All Since Citations h-index 30 17 iindex 62 Konstantinos Fokianos Professor of Statistics Verified email at lancaster. Michael H. View all. Professor of mathematics, University Cergy-Pontoise. Verified email at u-cergy. Articles Cited by Co-authors. Lecture Notes in Statistics.

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