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Functional Gaussian Approximation for Dependent Structures$
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Florence Merlevède, Magda Peligrad, and Sergey Utev

Print publication date: 2019

Print ISBN-13: 9780198826941

Published to Oxford Scholarship Online: April 2019

DOI: 10.1093/oso/9780198826941.001.0001

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Dependence Coefficients for Sequences

Dependence Coefficients for Sequences

(p.145) 5 Dependence Coefficients for Sequences
Functional Gaussian Approximation for Dependent Structures

Florence Merlevède

Magda Peligrad

Sergey Utev

Oxford University Press

In this chapter we survey several mixing conditions, which can be viewed as measures of departure from independence. We start with the traditional mixing coefficients such as the strong mixing coefficient and the coefficient of absolute regularity, as well as the ϕ‎- and ρ‎-mixing coefficients. We extend the definitions to sequences of random variables and give examples of such processes including classes of linear processes, Markov processes, and Gaussian processes. The most important property of these mixing coefficients is the fact that they allow the coupling with independent structures. This is the reason we pay special attention to the coupling properties of these mixing coefficients. The chapter continues with the presentation of weaker forms of mixing coefficients, defined by using smaller classes of functions. They allow us to enlarge the class of examples to more general functions of i.i.d. or to a larger class of dynamical systems.

Keywords:   mixing coefficients, coupling, weak dependence coefficients, examples of mixing processes, dynamical systems

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