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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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Introduction to Stochastic Processes

Introduction to Stochastic Processes

(p.1) 1 Introduction to Stochastic Processes
Functional Gaussian Approximation for Dependent Structures

Florence Merlevède

Magda Peligrad

Sergey Utev

Oxford University Press

We start by stating the need for a Gaussian approximation for dependent structures in the form of the central limit theorem (CLT) or of the functional CLT. To justify the need to quantify the dependence, we introduce illustrative examples: linear processes, functions of stationary sequences, recursive sequences, dynamical systems, additive functionals of Markov chains, and self-interactions. The limiting behavior of the associated partial sums can be handled with tools developed throughout the book. We also present basic notions for stationary sequences of random variables: various definitions and constructions, and definitions of ergodicity, projective decomposition, and spectral density. Special attention is given to dynamical systems, as many of our results also apply in this context. The chapter also surveys the basic theory of the convergence of stochastic processes in distribution, and introduces the reader to tightness, finite-dimensional convergence, and the need for maximal inequalities. It ends with the concepts of the moderate deviations principle and its functional form.

Keywords:   examples of stochastic processes, ergodicity, spectral density, convergence in distribution, maximal moment inequalities, moderate deviations principle

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