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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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Reversible Markov Chains

Reversible Markov Chains

Chapter:
(p.405) 14 Reversible Markov Chains
Source:
Functional Gaussian Approximation for Dependent Structures
Author(s):

Florence Merlevède

Magda Peligrad

Sergey Utev

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198826941.003.0014

This chapter is dedicated to the Gaussian approximation of a reversible Markov chain. Regarding this problem, the coefficients of dependence for reversible Markov chains are actually the covariances between the variables. We present here the traditional form of the martingale approximation including forward and backward martingale approximations. Special attention is given to maximal inequalities which are building blocks for the functional limit theorems. When the covariances are summable we present the functional central limit theorem under the standard normalization √n. When the variance of the partial sums are regularly varying with n, we present the functional CLT using as normalization the standard deviation of partial sums. Applications are given to the Metropolis–Hastings algorithm.

Keywords:   reversible Markov chains, maximal inequalities, forward martingale approximation, backward martingale approximation, functional central limit theorem

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