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: Autoregressive and moving average time series. Important properties such as causality. Autocovariance function and the spectral distribution of these models. Practical topics of time series like filtering and prediction. Basic concepts and definitions on the theory of stochastic processes, such as Wiener measure and process. General types of stochastic processes such as Gaussian, selfsimilar, compound and shot noise processes. Gaussian white noise, Langevin equation and Ornstein–Uhlenbeck process. Important related themes such as mean square properties of stationary processes and mean square integration. Spectral decomposition and spectral theorem of continuous time stationary processes. This central concept is followed by the theory of linear filters and their differential equations. 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.
Author(s): Riccardo Gatto
Series: World Scientific Series On Probability Theory And Its Applications, 4
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 414
City: Singapore
Contents
Preface
About the Author
List of Figures
List of Tables
1. Introduction
1.1 Stationary stochastic models and outline
1.2 Fourier analysis
1.2.1 Fourier series
1.2.2 Fourier transform
2. Stationary time series
2.1 Introduction
2.1.1 Stationarity and autocovariance function
2.1.2 Construction of stationary time series
2.1.3 Properties of the autocovariance function
2.2 ARMA time series
2.2.1 Definitions and motivations
2.2.2 Explicit representations of ARMA time series
2.2.3 Causality and invertibility
2.3 Autocovariance and related functions
2.3.1 Computation of autocovariance function
2.3.2 Partial autocorrelation function
2.3.3 Autocovariance generating function
2.4 Analysis in frequency domain
2.4.1 Complex-valued time series and spectral decomposition
2.4.2 Spectral distribution and Herglotz's theorem
2.4.3 Linear filters and spectral distribution of ARMA time series
2.4.4 AR and MA approximations to stationary time series
2.4.5 Periodogram and estimation of spectral density
2.5 Further classical topics on time series
2.5.1 Prediction with stationary time series
2.5.2 Estimation and asymptotic inference for mean and autocovariance function
2.5.3 Nonstationary ARIMA time series
2.5.4 Determination of order of AR time series
2.5.5 State space model and Kalman filter
2.5.6 Other stationary time series
3. Stationary processes with continuous time
3.1 Introduction
3.1.1 General definitions and theorems
3.1.2 Stationarity
3.1.3 Sample path regularity
3.2 Important stochastic processes
3.2.1 Gaussian processes
3.2.2 Wiener process
3.2.3 Selfsimilar processes and fractional Brownian motion
3.2.4 Counting processes
3.2.6 Point processes
3.2.7 Lévy processes
3.3 Mean square properties of stationary processes
3.4 Stochastic integrals
3.4.1 Two mean square integrals
3.4.2 Gaussian white noise
3.4.3 Ergodicity
3.4.4 Generalized and purely random stochastic processes
3.5 Spectral distribution and autocovariance function
3.5.1 Spectral distribution and Bochner's theorem
3.5.2 Inversion formulae
3.5.3 One-sided spectral distribution
3.5.4 Sampling and wrapping
3.6 Spectral decomposition of stationary processes and the spectral theorem
3.6.1 Discrete spectral process and distribution of a real process
3.6.2 Spectral decomposition
3.6.3 Discrete spectral process and distribution
3.7 Spectral analysis of Gaussian processes
3.7.1 Spectral decomposition of real Gaussian processes
3.7.2 Gaussian white noise and Ornstein-Uhlenbeck process
3.8 Spectral analysis of counting processes
3.9 Time invariant linear filters
3.9.1 Definitions
3.9.2 Some linear filters and their differential equations
3.9.3 Solutions of linear differential equations driven by stationary processes
3.9.4 Solutions of linear differential equations driven by white noise
3.9.5 Shot noise and filtered point process
4. Selected topics on stationary models
4.1 Stationary random fields
4.2 Circular time series
4.2.1 Circular correlation
4.2.2 Radially projected time series
4.2.3 Wrapped AR time series
4.2.4 Linked ARMA time series
4.2.5 Circular AR time series
4.2.6 vM time series
4.2.7 Model selection
4.3 Long range dependence
4.4 Nonintegrable spectral density and intrinsic stationarity
4.5 Unstable system
4.6 Hilbert transform and envelope
4.7 Simulation of stationary Gaussian processes
4.7.1 Simulation by Choleski factorization
4.7.2 Simulation by circulant embedding
4.7.3 Simulation by spectral decomposition
4.7.4 Simulation by ARMA approximation
4.8 Large deviations theory for time series
4.8.1 Some notions of large deviations theory
4.8.2 Large deviations approximation for AR(1) time series
4.9 Information theoretic results for time series
4.9.1 Circular distributions and information theory
4.9.2 The GvM and the Gaussian-GvM time series
Appendix A Mathematical complements
A.1 Hilbert space, L2 and Lp
A.1.1 Hilbert space
A.1.2 Space L2
A.1.3 Space Lp
A.2 Inequalities with random variables
A.3 Sequences of events and of random variables
A.4 Characteristic function
A.5 Theorems of integration
A.6 Remark on Riemann-Stieltjes integration
A.7 Taylor and Laurent series
A.8 Special functions and distributions
A.8.1 Function sinc
A.8.2 Dirac distribution and Dirac function
A.8.3 Gaussian or normal distribution
A.9 Linear differential equations
A.10 Fast Fourier transform
Appendix B Abbreviations, mathematical notation and data
B.1 Abbreviations
B.2 Mathematical notation
B.3 Data
Bibliography
Index
