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Bocconi & Springer #01: Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementationby Giovanni Peccati
Synopses & ReviewsPublisher Comments:The concept of Wiener chaos generalizes to an infinitedimensional 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 socalled 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 AuthorGiovanni 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.
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