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Bocconi & Springer #01: Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation

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Synopses & Reviews

Publisher Comments:

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 from Malliavin calculus to stochastic differential equations and from probabilistic approximations to mathematical finance. This book is concerned with combinatorial structures arising from the study of chaotic random variables related to infinitely divisible random measures. The combinatorial structures involved are those of partitions of finite sets, over which Möbius functions and related inversion formulae are defined. This combinatorial standpoint (which is originally due to Rota and Wallstrom) provides an ideal framework for diagrams, which are graphical devices used to compute moments and cumulants of random variables. Several applications are described, in particular, recent limit theorems for chaotic random variables. An Appendix presents a computer implementation in MATHEMATICA for many of the formulae.

Synopsis:

This is a book about combinatorial structures related to the law of random variables belonging to a so-called Wiener chaos, that is, to a space of multiple stochastic integrals (of a given fixed order) associated with some infinitely divisible random measure. The combinatorial structures involved are those of lattices of partitions of finite sets, over which we define incidence algebras, Mobius functions and associated inversion formulae. As discussed in the text, this combinatorial standpoint (due to Rota and Wallstrom) provides an ideal framework in order to systematically deal with diagram formulae, that are graphical devices used to compute moments and cumulants of random variables. This systematic study of diagram formulae, moments and cumulants is unique in its genre, and can be only found in our text. Several application are described, in particular to recent limit theorems for chaotic random variables. An explicit computer implementation into MATHEMATICA completes the work

About the Author

Giovanni Peccati is a Professor of Stochastic Analysis and Mathematical Finance at Luxembourg University. Murad S. Taqqu is a Professor of Mathematics and Statistics at Boston University.

Product Details

ISBN:
9788847016781
Author:
Peccati, Giovanni
Publisher:
Springer
Author:
Taqqu, Murad S.
Author:
Peccati
Subject:
Probability & Statistics - General
Subject:
Mathematical Analysis
Subject:
Combinatorics
Subject:
Statistics
Subject:
Diagram formulae
Subject:
Lattices of partitions
Subject:
Limit theorems
Subject:
Moments and cumulants
Subject:
Wiener chaos
Subject:
Probability Theory and Stochastic Processes
Subject:
Measure and Integration
Subject:
Measure and Integration A self-contained and probability-oriented introduction to the theory of lattice of partitions, with a unique software implementation that makes our book an ideal introduction to the field A complete and self-contained combinatorial
Subject:
Mathematics | Probability and Statistics
Subject:
Quantitative Finance
Copyright:
Edition Description:
2011
Series:
Bocconi & Springer Series
Series Volume:
1
Publication Date:
20110121
Binding:
HARDCOVER
Language:
English
Pages:
200
Dimensions:
235 x 155 mm

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Related Subjects

Health and Self-Help » Health and Medicine » Medical Specialties
Science and Mathematics » Mathematics » Analysis General
Science and Mathematics » Mathematics » Combinatorics
Science and Mathematics » Mathematics » Probability and Statistics » General
Science and Mathematics » Mathematics » Probability and Statistics » Probability Theory
Science and Mathematics » Mathematics » Probability and Statistics » Statistics

Bocconi & Springer #01: Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation New Hardcover
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Product details 200 pages Springer - English 9788847016781 Reviews:
"Synopsis" by , This is a book about combinatorial structures related to the law of random variables belonging to a so-called Wiener chaos, that is, to a space of multiple stochastic integrals (of a given fixed order) associated with some infinitely divisible random measure. The combinatorial structures involved are those of lattices of partitions of finite sets, over which we define incidence algebras, Mobius functions and associated inversion formulae. As discussed in the text, this combinatorial standpoint (due to Rota and Wallstrom) provides an ideal framework in order to systematically deal with diagram formulae, that are graphical devices used to compute moments and cumulants of random variables. This systematic study of diagram formulae, moments and cumulants is unique in its genre, and can be only found in our text. Several application are described, in particular to recent limit theorems for chaotic random variables. An explicit computer implementation into MATHEMATICA completes the work
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