Stella & Rose's Books

Front cover

Contents

• Preface
• 1 Preliminary Notions
• 1.1 Introduction
• 1.2 Sums of independent random variables: standardization
• 1.3 Moments, cumulants and their generating functions
• 1.4 Properties of sums of independent random variables
• 1.5 Calculation of moments and cumulants
• 1.6 Appendix: Orthogonal polynomials
• Further results and exercises
• Bibliographic notes
• 2 Some basic limiting procedures
• 2.1 Introduction
• 2.2 A simple example
• 2.3 Some convergence concepts
• 2.4 Convergence after functional transformation
• 2.5 Approximate linearization
• 2.6 Direct calculations with density functions and moment generating functions
• 2.7 Multidimensional version
• 2.8 Convergence of movements
• 2.9 Appendix: Some basic limit theorems
• Further results and exercises
• Bibliographic notes
• 3 Asymptotic expansions
• 3.1 Introduction
• 3.2 Integration by parts
• 3.3 Laplace expansion
• 3.4 Summation of series
• 3.5 Inversion of series
• 3.6 Asymptotic expansions of distributions by direct methods
• 3.7 Asymptotic expansions of distributions via generating functions
• 3.8 Asymptotic expansions for random variables
• 3.9 Asymptotic expansion with a second parameter
• Further results and exercises
• Bibliographic notes
• 4 Edgeworth and allied expansions
• 4.1 Introduction
• 4.2 Direct Edgeworth expansion
• 4.3 Tilted Edgeworth expansion
• 4.4 Cornish-Fisher of sums
• 4.5 Nonlinear function of sums
• Further results and exercises
• Bibliographic notes
• 5 Miscellany on multivariate distributions
• 5.1 Introduction
• 5.2 Special properties
• 5.3 Index notation
• 5.4 The exlog relations
• 5.5 Cumulants and moments
• 5.6 Cumulants of power series
• 5.7 Appendix: Tensorial Hermite polynomials
• 5.8 Appendix: Partially ordered sets, partitions and Mobius inversion
• 5.9 Appendix: Proof of the exlog relations
• Further results and exercises
• Bibliographic notes
• 6 Multivariate asymptotic expansions
• 6.1 Introduction
• 6.2 Laplace's method
• 6.3 Edgeworth expansions
• 6.4 Exponential methods
• 6.5 Tiled expansions
• 6.6 Large deviations
• 6.7 Mixed tilted-direct Edgeworth expansions
• 6.8 The delta method
• 6.9 Generalized formal Edgeworth expansions
• 6.10 Inversion
• 6.11 Appendix: Fourier transformation
• 6.12 Appendix: Laplace's method: regularity conditions
• 6.13 Appendix: Direct and tilted Edgeworth expansions: regularity conditions
• 6.14 Appendix: Legendre transformation
• Further results and exercises
• Bibliographic notes
• 7 Expansions for conditional distributions
• 7.1 Introduction
• 7.2 Direct - direct expansion
• 7.3 Expansions for condition cumulants
• 7.4 Tiled - tiled expansions
• 7.5 Mixed expansions
• Further results and exercises
• Bibliographic notes
• Postscript
• References
• Author Index
• Subject Index