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Fourier Analysis and Stochastic Processes
This work is unique as it provides a uniform treatment of the Fourier theories of functions (Fourier transforms and series, z-transforms), finite measures (characteristic functions, convergence in distribution), and stochastic processes (including arma series and point processes). It emphasises the links between these three themes. The chapter on the Fourier theory of point processes and signals structured by point processes is a novel addition to the literature on Fourier analysis of stochastic processes. It also connects the theory with recent lines of research such as biological spike signals and ultrawide-band communications. Although the treatment is mathematically rigorous, the convivial style makes the book accessible to a large audience. In particular, it will be interesting to anyone working in electrical engineering and communications, biology (point process signals) and econometrics (arma models). Each chapter has an exercise section, which makes Fourier Analysis and Stochastic Processes suitable for a graduate course in applied mathematics, as well as for self-study.
398 pages, Paperback
First published October 14, 2014
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July 9, 2025
Clearly, the theory of stochastic processes is indispensable to many domains of physics and electrical engineering, among other things. Unfortunately (as we have seen) Joseph L. Doob’s pioneering monograph on the subject from 1953 happens to be somewhat dated and unsuited for use as a textbook (we refer the reader to our review here). We wish to suggest instead Pierre Brémaud’s Fourier Analysis and Stochastic Processes, published by Springer-Verlag in 2014 as part of its Universitext series, as an up-to-date and ideal place to learn just about everything one could want to know about stochastic processes.
Brémaud’s elegant exposition is meant to be self-contained and all but presuppositionless, but is best approached after one has gained some experience with real analysis and probability theory at the graduate level. If one does have the requisite background, though, he will find the present work easy to follow for Brémaud tends to err on the side of full detail in his derivations and eschews a spare definition-theorem-proof style de rigueur at this level in favor of putting in enough pointers in introductory passages to orient the neophyte.
The first chapter contains a nice review of classical Fourier theory in L¹ resp. L². But, foreseeing their eventual employment in connection with stochastic processes, Brémaud dwells on several topics related to signals processing, such as discrete-time sampling, the z-transform and linear filtering, and the Shannon-Nyquist theorem. For someone in the position of this reviewer, for whom typical treatments of these things intended for electrical engineers can be frustrating due to their informal and heuristic attitude towards the mathematics, Brémaud’s is invaluable since he proceeds with perfect rigor and clarity. For this reason, it will be worth doing the homework exercises at the end of the chapter even for many who may already have been exposed to Fourier theory elsewhere. A good example of how Brémaud will illustrate his material with real-world use cases (unlike Stein-Shakarchi for the most part in their otherwise excellent introduction to Fourier analysis, our review here) would be the Nyquist condition for absence of intersymbol interference in pulse amplitude modulation.
The relatively brief second chapter connects Fourier analysis with probability theory. The concept of the characteristic function of a random variable crops up before, as an auxiliary to one of the standard approaches to proving the central limit theorem (due originally to Lyapunov), but has now a crucial role to play with regard to stochastic processes. For one wants if possible to decompose the stochastic process into its Fourier components, and, at least if it is wide-sense stationary, there exists a good sense according to which just this can be done. The key result in this context is Bochner’s theorem, which gives necessary and sufficient conditions under which a complex-valued function can be the Fourier transform of a positive finite measure. The basic idea behind its proof, drawing on the theorems of Lévy and Helley, is that one can correlate convergence of a sequence of characteristic functions with convergence of their corresponding measures. All of Brémaud’s proofs here continue to be admirably clear.
The heart of the book, however, is to be found in the extended chapter three on stochastic processes themselves. After a review which spells out the basic concepts, such as the meaning of wide-sense stationarity, and illustrates them with the Gaussian case (including Brownian motion, the Wiener process and the Wiener-Doob integral), the notion of power spectral measure is introduced. A wide-sense stationary process is essentially determined by is mean and covariance function and the latter is given uniquely as the Fourier transform of the power spectral measure. If one thinks of the stochastic process as happening in time, the power spectral measure will be a function of frequency. The concept of power spectral measure is nicely illustrated with a few examples of what one can do with it, for instance, define an Ornstein-Uhlenbeck process, a line spectrum, white noise, stable convolutional filtering, an approximate derivative and self-similarity of fractal Brownian motion.
What is remarkable is that one can not only characterize a wide-sense stationary process via its power spectral measure, one can also represent its trajectories directly in terms of a Fourier spectral decomposition. The usual Fourier transform does not exist in this case, but one can nevertheless obtain something analogous by means of the Doob integral. The corresponding representation is known as the Cramér-Khinchin decomposition. The remainder of the chapter takes up a handful of allied topics, such as random fields (a stochastic process in more than one independent variable), the Shannon-Nyquist theorem in this setting, linear composition of stochastic processes, the multivariate case and band-pass processes.
All of the above concerns processes in a continuous time parameter. In practice, however, one can know such a process only through measurements performed at discrete times, giving rise to what is called a time series. The machinery developed in chapter three has to be repeated in the case of time series, and this forms the topic of chapter four. After explaining how the basic notions may be recovered (the correlation function, the power spectral density, Cramér-Khinchin decomposition and Herglotz’ theorem as the discrete-time analogue of Bochner’s theorem), chapter four goes into a fair amount of detail on the important special case of autoregressive sequences and their prediction theory. The most interesting result in this setting is a generalization of prediction theory to what is called Wold’s decomposition, which expresses any wide-sense stationary time series in a unique manner as a sum of a purely deterministic part and a purely non-deterministic part. This raises an obvious question, namely, how well can one predict the future given a knowledge of the past? If a wide-sense stationary time series has non-zero deterministic components, one might hope to estimate these based on finitely many measurements in the past, where the accuracy of the estimate should improve as one increases the number of such measurements. This leads to the subject of so-called parametric spectral analysis, to which Brémaud devotes a rather complete discussion: the Levinson-Durbin algorithm, maximum entropy realization and Pisarenko’s realization.
The final chapter five covers point processes and their power spectra. A point process corresponds to a sequence of events where the interval in time between each successive event is a random variable. A good example would be the clicks registered by a Geiger counter. The process itself could then be pictured (formally) as a sum of Dirac delta functions. But since Dirac delta functions are not true functions, the ordinary concept of power spectral measure developed previously becomes inapplicable. The objective of this chapter is to develop a workable notion to put in its place, viz., the Bartlett spectral measure. To this reviewer, at any rate, chapter five seems to be more a matter of pushing around formalism for its own sake than conducive to any insight. For the latter, one might consult Feller’s first volume on probability theory, especially chapter eleven (our review here).
A word about the 105 homework exercises: none is at all hard, but they are definitely worth doing as they illustrate the theory nicely and often point out something one wouldn’t necessarily have thought of oneself, unless prompted to the realization by having to solve the problem. A good case in point would be a revealing problem in chapter two on Poisson’s law of rare events in the plane.
In sum, Brémaud's fine textbook on Fourier analysis of stochastic processes makes for a most pleasant learning experience!
Brémaud’s elegant exposition is meant to be self-contained and all but presuppositionless, but is best approached after one has gained some experience with real analysis and probability theory at the graduate level. If one does have the requisite background, though, he will find the present work easy to follow for Brémaud tends to err on the side of full detail in his derivations and eschews a spare definition-theorem-proof style de rigueur at this level in favor of putting in enough pointers in introductory passages to orient the neophyte.
The first chapter contains a nice review of classical Fourier theory in L¹ resp. L². But, foreseeing their eventual employment in connection with stochastic processes, Brémaud dwells on several topics related to signals processing, such as discrete-time sampling, the z-transform and linear filtering, and the Shannon-Nyquist theorem. For someone in the position of this reviewer, for whom typical treatments of these things intended for electrical engineers can be frustrating due to their informal and heuristic attitude towards the mathematics, Brémaud’s is invaluable since he proceeds with perfect rigor and clarity. For this reason, it will be worth doing the homework exercises at the end of the chapter even for many who may already have been exposed to Fourier theory elsewhere. A good example of how Brémaud will illustrate his material with real-world use cases (unlike Stein-Shakarchi for the most part in their otherwise excellent introduction to Fourier analysis, our review here) would be the Nyquist condition for absence of intersymbol interference in pulse amplitude modulation.
The relatively brief second chapter connects Fourier analysis with probability theory. The concept of the characteristic function of a random variable crops up before, as an auxiliary to one of the standard approaches to proving the central limit theorem (due originally to Lyapunov), but has now a crucial role to play with regard to stochastic processes. For one wants if possible to decompose the stochastic process into its Fourier components, and, at least if it is wide-sense stationary, there exists a good sense according to which just this can be done. The key result in this context is Bochner’s theorem, which gives necessary and sufficient conditions under which a complex-valued function can be the Fourier transform of a positive finite measure. The basic idea behind its proof, drawing on the theorems of Lévy and Helley, is that one can correlate convergence of a sequence of characteristic functions with convergence of their corresponding measures. All of Brémaud’s proofs here continue to be admirably clear.
The heart of the book, however, is to be found in the extended chapter three on stochastic processes themselves. After a review which spells out the basic concepts, such as the meaning of wide-sense stationarity, and illustrates them with the Gaussian case (including Brownian motion, the Wiener process and the Wiener-Doob integral), the notion of power spectral measure is introduced. A wide-sense stationary process is essentially determined by is mean and covariance function and the latter is given uniquely as the Fourier transform of the power spectral measure. If one thinks of the stochastic process as happening in time, the power spectral measure will be a function of frequency. The concept of power spectral measure is nicely illustrated with a few examples of what one can do with it, for instance, define an Ornstein-Uhlenbeck process, a line spectrum, white noise, stable convolutional filtering, an approximate derivative and self-similarity of fractal Brownian motion.
What is remarkable is that one can not only characterize a wide-sense stationary process via its power spectral measure, one can also represent its trajectories directly in terms of a Fourier spectral decomposition. The usual Fourier transform does not exist in this case, but one can nevertheless obtain something analogous by means of the Doob integral. The corresponding representation is known as the Cramér-Khinchin decomposition. The remainder of the chapter takes up a handful of allied topics, such as random fields (a stochastic process in more than one independent variable), the Shannon-Nyquist theorem in this setting, linear composition of stochastic processes, the multivariate case and band-pass processes.
All of the above concerns processes in a continuous time parameter. In practice, however, one can know such a process only through measurements performed at discrete times, giving rise to what is called a time series. The machinery developed in chapter three has to be repeated in the case of time series, and this forms the topic of chapter four. After explaining how the basic notions may be recovered (the correlation function, the power spectral density, Cramér-Khinchin decomposition and Herglotz’ theorem as the discrete-time analogue of Bochner’s theorem), chapter four goes into a fair amount of detail on the important special case of autoregressive sequences and their prediction theory. The most interesting result in this setting is a generalization of prediction theory to what is called Wold’s decomposition, which expresses any wide-sense stationary time series in a unique manner as a sum of a purely deterministic part and a purely non-deterministic part. This raises an obvious question, namely, how well can one predict the future given a knowledge of the past? If a wide-sense stationary time series has non-zero deterministic components, one might hope to estimate these based on finitely many measurements in the past, where the accuracy of the estimate should improve as one increases the number of such measurements. This leads to the subject of so-called parametric spectral analysis, to which Brémaud devotes a rather complete discussion: the Levinson-Durbin algorithm, maximum entropy realization and Pisarenko’s realization.
The final chapter five covers point processes and their power spectra. A point process corresponds to a sequence of events where the interval in time between each successive event is a random variable. A good example would be the clicks registered by a Geiger counter. The process itself could then be pictured (formally) as a sum of Dirac delta functions. But since Dirac delta functions are not true functions, the ordinary concept of power spectral measure developed previously becomes inapplicable. The objective of this chapter is to develop a workable notion to put in its place, viz., the Bartlett spectral measure. To this reviewer, at any rate, chapter five seems to be more a matter of pushing around formalism for its own sake than conducive to any insight. For the latter, one might consult Feller’s first volume on probability theory, especially chapter eleven (our review here).
A word about the 105 homework exercises: none is at all hard, but they are definitely worth doing as they illustrate the theory nicely and often point out something one wouldn’t necessarily have thought of oneself, unless prompted to the realization by having to solve the problem. A good case in point would be a revealing problem in chapter two on Poisson’s law of rare events in the plane.
In sum, Brémaud's fine textbook on Fourier analysis of stochastic processes makes for a most pleasant learning experience!
July 31, 2018
Maybe it is a great book, but French approach to textbooks is still alien to me.
Displaying 1 - 2 of 2 reviews