Fourier Analysis and Stochastic Processes

verfasst von: Pierre Brémaud

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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).

A careful review of the prerequisites (integration and probability theory in the appendix, Hilbert spaces in the first chapter) make the book self-contained. 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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Fourier Analysis of Functions

Abstract

The classical Fourier theory of functions is an indispensable prerequisite for the Fourier theory of stationary stochastic processes. By “classical” Fourier analysis, we mean Fourier series and Fourier transforms in \(L^1\) and \(L^2\), but also z-transforms which are the backbone of discrete-time signal processing together with the notion of (time- invariant) linear filtering. We spend some time with the famous Poisson summation formula—the bridge between Fourier transforms and Fourier series—which is intimately connected to the celebrated Shannon–Nyquist sampling theorem of signal processing and is of special interest to physicists and engineers in that it justifies the calculations involving the Dirac train of impulses without recourse to distribution theory. For the \(L^2\) Fourier theory of functions and sequences, some background in Hilbert spaces is required. The results obtained, such as the orthogonal projection theorem, the isometric extension theorem and the orthonormal basis theorem, will be recurrently used in the rest of the book.

Pierre Brémaud

Chapter 2. Fourier Theory of Probability Distributions

Abstract

Characteristic functions, that is Fourier transforms of probability measures, play a major role in Probability theory, in particular in the Fourier theory of wide-sense stationary stochastic processes, whose starting point is the notion of power spectral measure. It turns out that the existence of such a measure is a direct consequence of Bochner’s theorem of characterization of characteristic functions, and that the proof of its unicity is a rephrasing of Paul Lévy’s inversion theorem. Another result of Paul Lévy, characterizing convergence in distribution in terms of characteristic functions, intervenes in an essential way in the proof of Bochner’s theorem. In fact, characteristic functions are the link between the Fourier theory of deterministic functions and that of stochastic processes. This chapter could have been entitled “Convergence in distribution of random sequences”, a classical topic of probability theory. However, we shall need to go slightly beyond this and give the extension of Paul Lévy’s convergence theorem to sequences of finite measures (instead of probability distributions) as this is needed in Chap. 5 for the proof of existence of the Bartlett spectral measure.

Pierre Brémaud

Chapter 3. Fourier Analysis of Stochastic Processes

Abstract

This chapter is devoted to continuous-time wide-sense stationary stochastic processes and continuous parameter random fields. First, we introduce the notion of power spectral measure, that is, in a broad sense, the Fourier transform of the autocovariance function of such processes. (As we mentioned before, the existence and the unicity of the spectral measure are immediate consequences of Bochner’s theorem and of Paul Lévy’s inversion formula of the previous chapter.) Then, we look at the trajectories of the stochastic process themselves and give their Fourier decomposition. The classical Fourier theory of the first chapter does not apply there since the trajectories are in general not in \(L^1\) or \(L^2\). The corresponding result—Cramér–Khinchin’s decomposition—, is in terms of an integral of a special type, the Doob–Wiener integral of a function with respect to a stochastic process with uncorrelated increments, such as for instance the Wiener process, a particular case of Gaussian processes.

Pierre Brémaud

Chapter 4. Fourier Analysis of Time Series

Abstract

Discrete-time wide-sense stationary stochastic processes, also called time series, arise from discrete-time measurements (sampling) of random functions. A particularly mathematically tractable class of such processes consists of the so-called moving averages and auto-regressive (and more generally, arma) time series. This chapter begins with the general theory of wss discrete-time stochastic processes (which essentially reproduces that of wss continuous-time stochastic processes) and then gives the representation theory of arma processes, together with their prediction theory. The last section is concerned with the realization problem: what models fit a given finite segment of autocorrelation function of a time series? The corresponding theory is the basis of parametric spectral analysis.

Pierre Brémaud

Chapter 5. Power Spectra of Point Processes

Abstract

A simple point process on the positive half-line is, roughly speaking, a strictly increasing sequence \(\left\{ T_n\right\} _{n\in {\mathbb {N}}_+}\) of random variables taking their values in \([0, +\infty ]\) and called the event times.

Pierre Brémaud

Backmatter

Titel: Fourier Analysis and Stochastic Processes
verfasst von: Pierre Brémaud
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-09590-5
Print ISBN: 978-3-319-09589-9
DOI: https://doi.org/10.1007/978-3-319-09590-5