Fourier Analysis and Stochastic Processes

ISBN-13: 9783319095899 / Angielski / Miękka / 2014 / 385 str.

Pierre Bremaud

Opis

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.

Recenzje

“The main goal of this book is to define and study the Fourier transform of stochastic processes. … the author provides a modern unity of Fourier analysis and stochastic processes, presented together in a unique way. The ideal audience would be someone with exposure to analysis and probability at the graduate level who is interested in applications of Fourier analysis to stochastic processes. … this unique analytic treatment of stochastic processes provides a nice addition to the literature.” (Steven Michael Heilman, Mathematical Reviews, June, 2015)

“This is clearly an academic text, very well written and organized, with a high pedagogical quality … . This is a very interesting book adequate to support Master or PhD courses in Stochastic Processes. … it is accessible to larger audiences and useful for professionals working, for instance, in electrical engineering and communications, biology, economics and finance, but not only.” (Manuel Alberto M. Ferreira, Journal of Mathematics and Technology, Vol. 6 (1), 2015)

“This is a nice and modern book on the Fourier theory of functions, finite measures and stochastic processes with a lot of examples and exercises. … it may also be accessible to a large audience (electrical engineers, biologists and economists), since the author made the text as self contained as possible and provided a lot of examples in each chapter relevant from the point of view of applications. I warmly recommend the book for graduate students and researchers as well.” (Mátyás Barczy, Acta Scientiarum Mathematicarum, Vol. 81 (3-4), 2015)

Spis treści

Fourier analysis of functions.- Fourier theory of probability distributions.- Fourier analysis of stochastic processes.- Fourier analysis of time series.- Power spectra of point processes.

O autorze

Pierre Brémaud obtained his Doctorate in Mathematics from the University of Paris VI and his PhD from the department of Electrical Engineering and Computer Science of the University of California at Berkeley. He is a major contributor to the theory of stochastic processes and their applications, and has authored or co-authored several reference or textbooks on the subject.

Blurb

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.