1st Edition

# Stationary Stochastic Processes Theory and Applications

376 Pages 21 B/W Illustrations
by Chapman & Hall

375 Pages
by Chapman & Hall

Also available as eBook on:

Intended for a second course in stationary processes, Stationary Stochastic Processes: Theory and Applications presents the theory behind the field’s widely scattered applications in engineering and science. In addition, it reviews sample function properties and spectral representations for stationary processes and fields, including a portion on stationary point processes.

Features

• Presents and illustrates the fundamental correlation and spectral methods for stochastic processes and random fields
• Explains how the basic theory is used in special applications like detection theory and signal processing, spatial statistics, and reliability
• Motivates mathematical theory from a statistical model-building viewpoint
• Introduces a selection of special topics, including extreme value theory, filter theory, long-range dependence, and point processes
• Provides more than 100 exercises with hints to solutions and selected full solutions

This book covers key topics such as ergodicity, crossing problems, and extremes, and opens the doors to a selection of special topics, like extreme value theory, filter theory, long-range dependence, and point processes, and includes many exercises and examples to illustrate the theory. Precise in mathematical details without being pedantic, Stationary Stochastic Processes: Theory and Applications is for the student with some experience with stochastic processes and a desire for deeper understanding without getting bogged down in abstract mathematics.

Some Probability and Process Background
Sample space, sample function, and observables
Random variables and stochastic processes
Stationary processes and fields
Gaussian processes
Four historical landmarks

Sample Function Properties
Sample function continuity
Derivatives, tangents, and other characteristics
Stochastic integration
An ergodic result
Exercises

Spectral Representations
Complex-valued stochastic processes
Bochner’s theorem and the spectral distribution
Spectral representation of a stationary process
Gaussian processes
Stationary counting processes
Exercises

Linear Filters – General Properties
Linear time invariant filters
Linear filters and differential equations
White noise in linear systems
Long range dependence, non-integrable spectra, and unstable systems
The ARMA-family

Linear Filters – Special Topics
The Hilbert transform and the envelope
The sampling theorem
Karhunen-Loève expansion

Classical Ergodic Theory and Mixing
The basic ergodic theorem in L2
Stationarity and transformations
The ergodic theorem, transformation view
The ergodic theorem, process view
Ergodic Gaussian sequences and processes
Mixing and asymptotic independence

Vector Processes and Random Fields
Spectral representation for vector processes
Some random field theory
Exercises

Level Crossings and Excursions
Level crossings and Rice’s formula
Poisson character of high-level crossings
Marked crossings and biased sampling
The Slepian model
Crossing problems for vector processes and fields

A Some Probability Theory
Events, probabilities, and random variables
The axioms of probability
Expectations
Convergence
Characteristic functions
Hilbert space and random variables

B Spectral Simulation of Random Processes
The Fast Fourier Transform, FFT
Random phase and amplitude
Simulation scheme
Difficulties and details
Summary

C Commonly Used Spectra

D Solutions and Hints To Selected Exercises
Some probability and process background
Sample function properties
Spectral and other representations
Linear filters – general properties
Linear filters – special topics
Ergodic theory and mixing
Vector processes and random fields
Level crossings and excursions
Some probability theory

Bibliography
Index

### Biography

Georg Lindgren is with the Centre for Mathematical Sciences, Lund University, Sweden.

"I enjoyed reading this book. … It contains just about everything a graduate student would need to commence a Ph.D. in stochastic processes, or allied subjects. It is suitable not just for mathematically inclined statistics or probability students, but also for control engineering and signal processing students. It is an enjoyable read for established statisticians who would like a self-contained and easily read text on stochastic processes, continuous and discrete time, continuous and discrete state space. It is certainly an easier read than Doob (1953)! … I would also recommend this book to any colleague wanting to learn a lot about stationary processes in a very short period of time."
—Barry Quinn, Macquarie University, Sydney, Australia, from the Australian & New Zealand Journal of Statistics, Volume 57, 2015

"This text provides an excellent exposition of advanced theory and applications for stationary stochastic processes. The author has gathered an extensive amount of methodology and recent research that makes this book a natural successor to the classic books by Cramer and Leadbetter (1967) and Yaglom (1987). What is of particular importance is that most of the exposition and proofs do not involve advanced probability and measure theory. Consequently, the concepts are more accessible to students and researchers with a basic background in probability theory and stochastic processes, and therefore it can be used as a second course in stationary processes in mathematics and statistics, as well as engineering. … the topics covered should make this book very attractive to the active researcher using Fourier and time series analysis methods. Moreover, this text could serve as a natural companion to recent texts on spatial and spatio-temporal approaches (such as Cressie and Wikle, 2011), by providing the theoretical background on some of these methods."
Journal of the American Statistical Association, June 2014

"This is a very nice book about second-order stationary stochastic processes for readers who have already some knowledge about stochastic processes and want to deepen and extend their knowledge in this field. … the book is not too dry and theory is always motivated by interesting examples coming from engineering and the physical sciences. This book is in my opinion also a very good basis for an advanced course on this topic as it is logically structured and as each chapter also contains many exercises."
—Klaus Nordhausen, International Statistical Review (2013), 81

"Stationary Stochastic Processes manages to present a wide topic of applied mathematics and does not fall off from the thin ridge that lies between the probabilistic and the more signal process (deterministic) representation of stationary processes. A lot of material can be found therein, and it will be very helpful to young researchers."
—Marc Hoffmann, Université Paris–Dauphine, CHANCE, 26.3

"This book offers quite a unique approach and selection of topics within the modern field of stochastic processes. It aims at providing theoretical insight to graduate students and researchers in engineering and science … also those coming from the theoretical side who want to know more about the applications will benefit from this book. A common theme for the book is the bridge-building between different audiences. … Without being mathematically over-demanding, the book builds up the relevant theory in a very intuitive yet rigorous way that helps the reader to a deeper understanding of definitions and results that could otherwise be mystifying. This intuition-driven approach also provides a common thread throughout the book."
—Claudia Klüppelberg and Morten Grud Rasmussen, Technische Universität München, Germany

"In the same vein as the classic works by Cramér and Leadbetter and Yaglom, Lindgren’s book offers both an introduction for graduate students and valuable insights for established researchers into the theory of stationary processes and its application in the engineering and physical sciences. His approach is rigorous but with more focus on the big picture than on detailed mathematical proofs. Strong points of the book are its coverage of ergodic theory, spectral representations for continuous- and discrete-time stationary processes, basic linear filtering, the Karhunen–Loève expansion, and zero crossings. While the text is mainly focused on one-dimensional processes, there is also coverage of vector-valued processes and random fields. Particularly appealing features of the book are its numerous examples and remarks (some providing interesting historical background). The structure of the book is such that it can be recommended both as a classroom text and for individual study."
—Don Percival, Senior Principal Mathematician, University of Washington, Seattle, USA

"From my start as a graduate student in probability and statistics, Stationary and Related Stochastic Processes by Cramer and Leadbetter has always been one of my favorites. … In many respects, Lindgren’s Stationary Stochastic Processes: Theory and Applications is an updated and expanded version that has captured much of the same spirit (and topics!) as the Cramer and Leadbetter classic. While there have been a number of new and good books published recently on spatial statistics, none cover some of the key important topics such as sample path properties and level crossings in a comprehensive and understandable fashion like Lindgren’s book. This book is required reading for all of my PhD students working in spatial statistics and related areas."
—Richard A. Davis, Howard Levene Professor of Statistics, Columbia University, New York, USA

"Georg Lindgren's new book is a most attractive presentation of the theory and application of these processes, with an emphasis on second order properties and Fourier methods. The theory is described with a view towards application but the treatment does not duck the technical mathematics; rather it presents, honestly and clearly, all mathematical ideas that are needed, accompanying them by motivation and interpretation that keep the wider purpose in mind. … The choice of topics is consistent with the general approach: interesting, engaging, giving the reader ample evidence of the utility of the theory and showing how practical needs drive it forward. … the book is authoritative and stimulating, a worthy champion of the tradition of Cramer and Leadbetter admired by the author (and many others). It is a rich, inspiring book, full of good sense and clarity, an outstanding text in this important field."
—Clive Anderson, University of Sheffield, UK

"It is with a great pleasure I welcome this book by Prof. G. Lindgren. At first thought, it seems that such a book is redundant as this is a classical theme well covered by classical texts. But it is not! This book is a unique blend of classical theory and application theory —where stochastic processes theory meets applications in engineering and science. I particular enjoyed the chapter about level crossings and excursions … . The book is very well written, the themes are well chosen and the style is relaxed but precise without being pedantic. My only regret is that this book did not appear earlier! This book is highly recommended!"
—Håvard Rue, Norwegian University of Science and Technology, Trondheim