Wiener Chaos: Moments, Cumulants and Diagrams A survey with computer
The concept of Wiener chaos generalizes to an infinite-dimensional setting the properties of orthogonal polynomials associated with probability distributions on the real line. It plays a crucial role in modern probability theory, with applications ranging
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Giovanni Peccati • Murad S. Taqqu
Wiener Chaos: Moments, Cumulants and Diagrams A survey with computer implementation
Giovanni Peccati Mathematics Research Unit University of Luxembourg Luxembourg [email protected]
Murad S. Taqqu Department of Mathematics and Statistics Boston University Boston [email protected]
Additional material to this book can be downloaded from http://extras.springer.com Password: 978-88-470-1678-1 B&SS – Bocconi & Springer Series ISSN print edition: 2039-1471
ISSN electronic edition : 2039-148X
ISBN 978-88-470-1678-1
e-ISBN 978-88-470-1679-8
DOI 10.1007/978-88-470-1679-8
Library of Congress Control Number: 2010929482
Springer Milan Dordrecht Heidelberg London New York © Springer-Verlag Italia 2011
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XI
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 5
2
The lattice of partitions of a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Partitions of a positive integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Partitions of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Partitions of a set and partitions of an integer . . . . . . . . . . . . . . . . . . . 2.4 Bell polynomials, Stirling numbers and Touchard polynomials . . . . 2.5 Möbius functions and Möbius inversion on partitions . . . . . . . . . . . . 2.6 Möbius functions on partially ordered sets . . . . . . . . . . . . . . . . . . . . . . 2.
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