Lukacs E. Characteristic functions

London: Griffin, 1970. — 360 p.This volume studies characteristic functions--which play an essential role in probability and statistics-- for their intrinsic, mathematical interest.Preface to Second Edition
Preface to First Edition
Distribution functions
Examples of distribution functions
The method of integral transforms
Moments
Elementary properties of characteristic functions
Lebesgue decomposition of characteristic functions
Characteristic functions and moments
The second characteristic
The uniqueness theorem
Inversion formulae
The convolution theorem
Limits of distribution functions
The theorems of Helly
The continuity theorem
Infinite convolutions
Necessary conditions
Necessary and sufficient conditions
Sufficient conditions
Supplementary remarks concerning non-negative definite functions
Unimodal distributions
An essential property of characteristic functions
Preliminary remarks on factorizations
Definition of infinitely divisible characteristic functions
Elementary properties of infinitely divisible characteristic functions
Construction of infinitely divisible characteristic functions
Canonical representations
A limit theorem
Characteristic functions of stable distributions
Frequency functions of stable distributions
Asymptotic expansions and integral representations of stable densities
Unimodality of stable distributions
Self-decomposnble distributions
Some notations and lemmas
General decomposition theorems
Indecomposable characteristic functions
The strip of regularity and the integral representation
Analytic characteristic functions and their distribution functions
Criteria for analytic characteristic functions
Periodic analytic characteristic functions
Analytic characteristic functions as solutions of certain differential equations
Properties of the factors of an analytic characteristic function
Factorization of certain entire characteristic functions
Determination of certain entire characteristic functions by properties of their factors
Infinitely divisible analytic characteristic functions
The class L
A sufficient condition for membership of I_
A necessary condition for membership of I_
Infinitely divisible characteristic functions with bounded Poisson spectrum
Theorems concerning certain factorizations
General theorems on \alpha-decompositions of analytic characteristic functions
Special results concerning (X-decompositions
The integral representation
Infinitely divisible boundary characteristic functions
Mixtures of distribution functions
Transformations of characteristic functions
Appendix A: The notations O and o
Appendix C: Weierstrass' approximation theorem
Appendix D: Order and type of entire functions
Appendix E: Proof of lemmas needed in Chapter
Appendix F: Schwarz's reflection principle
List of Examples of Characteristic Functions