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    Wiener Chaos - Moments, Cumulants and Diagrams A Survey with Computer Implementation

    ISBN-10: 8847016789
    ISBN-13: 9788847016781
    Author(s): Giovanni Peccati, Murad S. Taqqu
    List price: $139.00
    Buy it from: $108.37
    Rent it from: $59.82
    This item will ship on Friday, December 26 .

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    List Price: $139.00
    Publisher: Springer
    Binding: Hardcover
    Pages: 274
    Size: 6.50" wide x 9.25" long x 0.25" tall
    Weight: 1.540
    Language: English

    Giovanni Peccati is Full Professor in Stochastic Analysis at the University of Luxembourg.

    Preface
    Introduction
    Overview
    Some related topics
    The lattice of partitions of a finite set
    Partitions of a positive integer
    Partitions of a set
    Partitions of a set and partitions of an integer
    Bell polynomials, Stirling numbers and Touchard polynomials
    M�bius functions and M�bius inversion on partitions
    M�bius functions on partially ordered sets
    Incidence algebras and general M�bius inversion
    Computing a M�bius function
    Further properties of M�bius functions and a proof of (2.5.21)
    Combinatorial expressions of cumulants and moments
    Cumulants
    Relations involving moments and cumulants
    Bell and Touchard polynomials and the Poisson distribution
    Diagrams and multigraphs
    Diagrams
    Solving the equation � � � = 0
    From Gaussian diagrams to multigraphs
    Summarizing diagrams and multigraphs
    Wiener-It� integrals and Wiener chaos
    Completely random measures
    Single Wiener-It� integrals
    Infinite divisibility of single integrals
    Multiple stochastic integrals of elementary functions
    Wiener-It� stochastic integrals
    Integral notation
    The role of diagonal measures
    Example: completely random measures on the real line
    Chaotic representation
    Computational rules
    Multiple Gaussian integrals of elementary functions
    Multiple Poisson integrals of elementary functions
    A stochastic Fubini theorem
    Multiplication formulae
    The general case
    Contractions
    Symmetrization of contractions
    The product of two integrals in the Gaussian case
    The product of two integrals in the Poisson case
    Diagram formulae
    Formulae for moments and cumulants
    MSets and MZeroSets
    The Gaussian case
    The Poisson case
    From Gaussian measures to isonormal Gaussian processes
    Multiple stochastic integrals as Hermite polynomials
    Chaotic decompositions
    Isonormal Gaussian processes
    Wiener chaos
    Contractions, products and some explicit formulae
    Hermitian random measures and spectral representations
    The isonormal approach
    The Gaussian measure approach
    Spectral representation
    Stochastic processes
    Hermite processes
    Further properties of Hermitian Gaussian measures
    Caveat about normalizations
    Some facts about Charlier polynomials
    Limit theorems on the Gaussian Wiener chaos
    Some features of the laws of chaotic random variables (Gaussian case)
    A useful estimate
    A general problem
    One-dimensional CLTs in the Gaussian case
    An application to Brownian quadratic functionals
    Proof of the propositions
    Further combinatorial implications
    A multidimensional CLT
    CLTs in the Poisson case: the case of double integrals
    Practical implementation using Mathematica
    General introduction
    When Starting
    Integer Partitions
    IntegerPartitions
    IntegerPartitionsExponentRepresentation
    Basic Set Partitions
    SingletonPartition
    MaximalPartition
    SetPartitions
    PiStar
    Operations with Set Partitions
    PartitionIntersection
    PartitionUnion
    MeetSolve
    JoinSolve
    JoinSolveGrid
    MeetAndJoinSolve
    CoarserSetPartitionQ
    CoarserThan
    Partition Segments
    PartitionSegment
    ClassSegment
    ClassSegmentSolve
    SquareChoose
    Bell and Touchard polynomials
    StirlingS2
    BellPolynomialPartial
    BellPolynomialPartialAuto
    BellPolynomialComplete
    BellPolynomialCompleteAuto
    BellB
    TouchardPolynomial
    TouchardPolynomialTable
    TouchardPolynomialCentered
    TouchardPolynomialCenteredTable
    Mobius Formula
    MobiusFunction
    MobiusRecursionCheck
    MobiusInversionFormulaZero
    MobiusInversionFormulaOne
    Moments and Cumulants
    MomentToCumulants
    MomentToCumulantsBell
    MomentToCumulantsBellTable
    CumulantToMoments
    CumulantToMomentsBell
    CumulantToMomentsBellTable
    MomentProductToCumulants
    CumulantVectorToMoments
    CumulantProductVectorToCumulantVectors
    GridCumulantProductVectorToCumulantVectors
    CumulantProductVectorToMoments
    Gaussian Multiple Integrals
    GaussianIntegral
    Poisson Multiple Integrals
    BOne
    BTwo
    PBTwo
    PoissonIntegral
    PoissonIntegralExceed
    Contraction and Symmetrization
    ContractionWithSymmetrization
    ContractionIntegration
    ContractionIntegrationWithSymmetrization
    Solving Partition Equations Involving �*
    PiStarMeetSolve
    PiStarJoinSolve
    PiStarMeetAndJoinSolve
    Product of Two Gaussian Multiple Integrals
    ProducfTwoGaussianIntegrals
    Product of Two Poisson Multiple Integrals
    ProductTwoPoissonIntegrals
    MSets and MZeroSets
    MSets
    MSetsEqualTwo
    MSetsGreaterEqualTwo
    MZeroSets
    MZeroSetsEqualTwo
    MZeroSetsGreaterEqualTwo
    Hermite Polynomials
    HermiteRho
    HermiteRhoGrid
    Hermite
    HermiteGrid
    HermiteH
    HermiteHGrid
    Poisson-Charlier Polynomials
    Charlier
    CharlierGrid
    CharlierCentered
    CharlierCenteredGrid
    Tables of moments and cumulants
    Centered and scaled Cumulant to Moments
    Centered Cumulant to Moments
    Cumulant to Moments
    Centered and scaled Moment to Cumulants
    Centered Moment to Cumulants
    Moment to Cumulants
    References
    Index

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