Wiener Chaos: Moments, Cumulants and Diagrams [electronic resource] : A survey with Computer Implementation /
Peccati, Giovanni.author.
Taqqu, Murad S.author.
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.
Description based upon print version of record.
Title Page; Copyright Page; Table of Contents; Preface; 1 Introduction; 1.1 Overview; 1.2 Some related topics; 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; 2.6.1 Incidence algebras and general Möbius inversion; 2.6.2 Computing a Möbius function
5.4 Multiple stochastic integrals of elementary functions5.5 Wiener-Itô stochastic integrals; 5.6 Integral notation; 5.7 The role of diagonal measures; 5.8 Example: completely random measures on the real line; 5.9 Chaotic representation; 5.10 Computational rules; 5.11 Multiple Gaussian integrals of elementary functions; 5.12 Multiple Poisson integrals of elementary functions; 5.13 A stochastic Fubini theorem; 6 Multiplication formulae; 6.1 The general case; 6.2 Contractions; 6.3 Symmetrization of contractions; 6.4 The product of two integrals in the Gaussian case
6.5 The product of two integrals in the Poisson case7 Diagram formulae; 7.1 Formulae for moments and cumulants; 7.2 MSets and MZeroSets; 7.3 The Gaussian case; 7.4 The Poisson case; 8 From Gaussian measures to isonormal Gaussian processes; 8.1 Multiple stochastic integrals as Hermite polynomials; 8.2 Chaotic decompositions; 8.3 Isonormal Gaussian processes; 8.4 Wiener chaos; 8.5 Contractions, products and some explicit formulae; 9 Hermitian random measuresand spectral representations; 9.1 The isonormal approach; 9.2 The Gaussian measure approach; 9.3 Spectral representation
9.4 Stochastic processes9.5 Hermite processes; 9.6 Further properties of Hermitian Gaussian measures; 9.7 Caveat about normalizations; 10 Some facts about Charlier polynomials; 11 Limit theorems on the Gaussian Wiener chaos; 11.1 Some features of the laws of chaotic random variables (Gaussian case); 11.2 A useful estimate; 11.3 A general problem; 11.4 One-dimensional CLTs in the Gaussian case; 11.5 An application to Brownian quadratic functionals; 11.6 Proof of the propositions; 11.7 Further combinatorial implications; 11.8 A multidimensional CLT
12 CLTs in the Poisson case:the case of double integrals
Includes bibliographical references and index.
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.
Probabilities.
Economics, Mathematical .
Combinatorics.
Measure theory.
Probability Theory and Stochastic Processes.
Quantitative Finance.
Combinatorics.
Measure and Integration.
Bocconi & Springer Series