Stochastic Models for Time Series

  • Paul Doukhan

Part of the Mathématiques et Applications book series (MATHAPPLIC, volume 80)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Independence and Stationarity

    1. Front Matter
      Pages 1-1
    2. Paul Doukhan
      Pages 3-8
    3. Paul Doukhan
      Pages 9-26
    4. Paul Doukhan
      Pages 27-48
    5. Paul Doukhan
      Pages 49-70
  3. Models of Time Series

    1. Front Matter
      Pages 71-71
    2. Paul Doukhan
      Pages 73-99
    3. Paul Doukhan
      Pages 101-114
    4. Paul Doukhan
      Pages 115-166
    5. Paul Doukhan
      Pages 167-173
  4. Dependence

    1. Front Matter
      Pages 175-175
    2. Paul Doukhan
      Pages 177-188
    3. Paul Doukhan
      Pages 189-204
    4. Paul Doukhan
      Pages 205-224
    5. Paul Doukhan
      Pages 225-246
  5. Back Matter
    Pages 247-308

About this book

Introduction

This book presents essential tools for modelling non-linear time series. The first part of the book describes the main standard tools of probability and statistics that directly apply to the time series context to obtain a wide range of modelling possibilities. Functional estimation and bootstrap are discussed, and stationarity is reviewed. The second part describes a number of tools from Gaussian chaos and proposes a tour of linear time series models. It goes on to address nonlinearity from polynomial or chaotic models for which explicit expansions are available, then turns to Markov and non-Markov linear models and discusses Bernoulli shifts time series models. Finally, the volume focuses on the limit theory, starting with the ergodic theorem, which is seen as the first step for statistics of time series. It defines the distributional range to obtain generic tools for limit theory under long or short-range dependences (LRD/SRD) and explains examples of LRD behaviours. More general techniques (central limit theorems) are described under SRD; mixing and weak dependence are also reviewed. In closing, it describes moment techniques together with their relations to cumulant sums as well as an application to kernel type estimation.The appendix reviews basic probability theory facts and discusses useful laws stemming from the Gaussian laws as well as the basic principles of probability, and is completed by R-scripts used for the figures. Richly illustrated with examples and simulations, the book is recommended for advanced master courses for mathematicians just entering the field of time series, and statisticians who want more mathematical insights into the background of non-linear time series.

 

Keywords

60G10 37M10 32A25 60F05 60F15 60G18 60G12 60J05 62J12 62M10 62M15 91B84 Non-linear time series Integer valued models Markov chains Stochastic processes Gaussian convergence Spectral estimation Memory models LARCH-type models Weak dependence conditions Functional estimation Bootstrap Non-Markove linear models

Authors and affiliations

  • Paul Doukhan
    • 1
  1. 1.Laboratory of MathematicsUniversity Cergy-PontoiseCergy-PontoiseFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-76938-7
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-76937-0
  • Online ISBN 978-3-319-76938-7
  • Series Print ISSN 1154-483X
  • Series Online ISSN 2198-3275
  • About this book
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