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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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Application to the Uniform Laws of Large Numbers for Dependent Processes

Application to the Uniform Laws of Large Numbers for Dependent Processes

Chapter:
(p.438) 16 Application to the Uniform Laws of Large Numbers for Dependent Processes
Source:
Functional Gaussian Approximation for Dependent Structures
Author(s):

Florence Merlevède

Magda Peligrad

Sergey Utev

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198826941.003.0016

As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling of the initial sequence by an independent one having the same marginal. In this chapter, we shall see how the coupling results mentioned in Section 5.1.3 are very useful to derive uniform laws of large numbers for mixing sequences. The uniform laws of large numbers extend the classical laws of large numbers from a single function to a collection of such functions. We shall address this question for sequences of random variables that are either absolutely regular, or ϕ‎-mixing, or strongly mixing. In all the obtained results, no condition is imposed on the rates of convergence to zero of the mixing coefficients.

Keywords:   uniform laws of large numbers, Glivenko–Cantelli class, coupling, mixing sequences

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