Fourier Analysis and Stochastic Processes

431 Pages·2014·17.56 MB·English

by  Pierre Brémaud

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Pierre Brémaud
Fourier Analysis and Stochastic
Processes
Pierre Brémaud
Inria, Paris, France
École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
ISSN 0172-5939 e-ISSN 2191-6675
ISBN 978-3-319-09589-9 e-ISBN 978-3-319-09590-5
DOI 10.1007/978-3-319-09590-5
Springer Cham Heidelberg New York Dordrecht London Library of Congress
Control Number: 2014946193
Mathematics Subject Classification (2010): 42A38, 42B10, 60G10, 60G12,
60G35, 60G55
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Pour Marion
Introduction
Pierre Brémaud
A unified treatment of all the aspects of Fourier theory relevant to the Fourier
analysis of stochastic processes is not only unavoidable, but also intellectually
satisfying, and in fact time-saving. This is why this book contains the classical
Fourier theory of functions (Fourier series and transforms in L 1 and L 2 , z-
transforms), the Fourier theory of probability distributions (characteristic
functions, convergence in distribution) and the elements of Hilbert space theory
(orthogonal projection principle, orthonormal bases, isometric extension
theorem) which are indispensable prerequisites for the Fourier analysis of
stochastic processes (random fields, time series and point processes) of the last
three chapters.
Stochastic processes and random fields are studied from the point of view of
their second order properties embodied in two functions: the mean function and
the covariance function (or the covariance measure in the case of point process
random fields). This limited information suffices to satisfy most demands of
signal processing and has made this class of processes a rich source of models:
white and coloured noises, series and, more recently, complex signals
ARMA
based on point processes. Wide-sense stationary processes are, in addition to
their usefulness, supported by an elegant theory at the interface of Fourier
analysis and Hilbert spaces on one hand, and probability and stochastic
processes on the other hand. This book presents the classical results in the field
together with more recent ones on the power spectral measure (or Bartlett
spectrum) of wide-sense stationary point processes and related stochastic
processes, such as (non-Poissonian) shot noises, modulated point processes,
random samples, etc.
The mathematical prerequisites on integration and probability theory are
reviewed in the Appendix, generally without proof, except for the completeness
of L p -spaces (stating in particular that the square-integrable functions form a
Hilbert space), because this result is the foundation of the second-order theory of
stochastic processes, together with some results of Hilbert space theory. The
latter are given, this time with proofs, in the course of the Chap. 1 . No
prerequisite on stochastic processes is necessary, since the relevant models
(especially Gaussian processes and Brownian motion, but also Markov chains,
Poisson processes and renewal processes) are presented in detail in the main text
as the need arises.
The Chap. 1 gives the basic Fourier theory of functions. The first section
deals with the Fourier theory in L 1 , featuring among other topics the Poisson
summation formula and the Shannon–Nyquist sampling theorem. The second
section gives the theory of the z-transform, which is essential to the Fourier
theory of time series in Chap. 4 . The third section provides, in a self-contained
way, the background in Hilbert spaces which is a prerequisite for the Fourier
theory in L 2 of the last section, as well as for the rest of the book.
The Chap. 2 provides the interface between the Fourier analysis of functions
and the Fourier analysis of stochastic processes, namely the theory of
characteristic functions and the pivotal Bochner theorem, which guarantees
under very general conditions the existence and uniqueness of the power spectral
measure of wide-sense stationary stochastic processes. This theorem relies on
Paul Lévy’s theorem of characterization of convergence of probability
distributions. A slight extension of the latter to the problem of convergence of
finite measures is given since it will be needed in the proof of existence of the
power spectral measure, or Bartlett spectrum, of point processes.
Chapters 3 – 5 concern stochastic processes. There are two types of
continuous-time stochastic processes and continuous-space random fields that
may be distinguished, as we do in the present text. The classical one and indeed
most frequently—if not exclusively—dealt with in textbooks, whether applied or
theoretical, concerns mathematical objects, such as Gaussian processes, for
which the second-order analysis, in particular the spectral representation theory,
is reasonably well-developed and can serve as a starting point for their spectral
analysis. The second one concerns point processes. These are underlying many
models of the first type (regenerative processes among others) and are in general
not studied per se. However, the analysis of signals whose point process
structure is essential, such as those arising in biology (the spike trains across the
nervous fibers) and in communications (the so-called ultra-wide-band signals in
which the information is coded into a sequence of random times) require a
special treatment. Chapter 3 concerns the first category of signals, whereas
Chap. 5 deals with stochastic processes structured by point processes. Chapter
4 is devoted to discrete-time wide-sense stationary processes, with emphasis on
the models, of interest in signal processing as well as in econometrics,
ARMA
among other fields of application.
The formulation of the mathematical theory, the choice of topics and of
examples should allow students and researchers in the applied sciences to
recognize the objects and paradigms of their respective trade, the aim being to
give a firm mathematical basis on which to develop applications-oriented
research. In particular, this book features examples relevant to signal processing
and communications: Shannon–Nyquist sampling theorem, transfer functions,
white noise, pulse-amplitude modulation, filtering, narrowband signals,
sampling in the presence of jitter, spike trains, ultra-wide-band signals, etc.
However, this choice of examples in one of the richest domains of application of
Fourier theory does not diminish the generality of the mathematical theory
developed in the text. It will perhaps remind the reader that Fourier theory was
originally a physical theory.
I would like to close this introduction with the expression of my sincere
gratitude to Justin Salez who reviewed and corrected important portions of the
manuscript.
Paris, March 2014
Contents
1 Fourier Analysis of Functions
1.1 Fourier Theory in
1.1.1 Fourier Transform and Fourier Series
1.1.2 Convergence Theory
1.1.3 The Poisson Summation Formula
1.2 Z-Transforms
1.2.1 Fourier Sums
1.2.2 Transfer Functions
1.3 Hilbert Spaces
1.3.1 Isometric Extension
1.3.2 Orthogonal Projection
1.3.3 Orthonormal Expansions
1.4 Fourier Theory in
1.4.1 Fourier Transform in
1.4.2 Fourier Series in
1.5 Exercises
2 Fourier Theory of Probability Distributions
2.1 Characteristic Functions
2.1.1 Basic Facts