Stationary Stochastic Models

This volume provides a unified mathematical introduction to stationary time series models and to continuous time stationary stochastic processes. The analysis of these stationary models is carried out in time domain and in frequency domain. It begins with a practical discussion on stationarity, by which practical methods for obtaining stationary data are described. The presented topics are illustrated by numerous examples. Readers will find the following covered in a comprehensive manner:

At the end, some selected topics such as stationary random fields, simulation of Gaussian stationary processes, time series for planar directions, large deviations approximations and results of information theory are presented. A detailed appendix containing complementary materials will assist the reader with many technical aspects of the book.

Contents:

• Introduction:

  • Stationary Stochastic Models and Outline
  • Fourier Analysis

• Stationary Time Series:

  • Introduction
  • ARMA Time Series
  • Autocovariance and Related Functions
  • Analysis in Frequency Domain
  • Further Classical Topics on Time Series

• Stationary Processes with Continuous Time:

  • Introduction
  • Important Stochastic Processes
  • Mean Square Properties of Stationary Processes
  • Stochastic Integrals
  • Spectral Distribution and Autocovariance Function
  • Spectral Decomposition of Stationary Processes and the Spectral Theorem
  • Spectral Analysis of Gaussian Processes
  • Spectral Analysis of Counting Processes
  • Time Invariant Linear Filters

• Selected Topics on Stationary Models:

  • Stationary Random Fields
  • Circular Time Series
  • Long Range Dependence
  • Nonintegrable Spectral Density and Intrinsic Stationarity
  • Unstable System
  • Hilbert Transform and Envelope
  • Simulation of Stationary Gaussian Processes
  • Large Deviations Theory for Time Series
  • Information Theoretic Results for Time Series

• Appendices:

  • Mathematical Complements
  • Abbreviations, Mathematical Notation and Data

Readership: Upper-level undergraduate and graduate students, for lectures on time series or on stochastic processes with continuous time. Researchers in academia and applied scientists in the industry, in the field of time series or stationary processes. These lectures can be given to students of mathematics or statistics as well as to students from other technical fields, at Bachelor's upper-level and at Master's level.
Key Features:

• The topics are presented progressively, by going from discrete to continuous time, by studying connected questions and by completing abstract theory with compelling examples
• Suited for upper-level undergraduate and graduate students, researchers in academia and scientists in industry
• Covers both fundamentals and practical aspects of stationary models
• This book provides a unified presentation of stationary time series and continuous time stationary processes
• The appendix provides complementary material that assists the reader with technical aspects of the book

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