Characteristic functions

This volume studies characteristic functions--which play an essential role in probability and statistics-- for their intrinsic, mathematical interest. Title......Page 2 Preface to Second Edition......Page 3 Preface to First Edition......Page 6 Contents......Page 8 1.1 Distribution functions......Page 10 1.2 Examples of distribution functions......Page 14 1.3 The method of integral transforms......Page 18 1.4 Moments......Page 20 2.1 Elementary properties of characteristic functions......Page 24 2.2 Lebesgue decomposition of characteristic functions......Page 28 2.3 Characteristic functions and moments......Page 29 2.4 The second characteristic......Page 35 3.1 The uniqueness theorem......Page 37 3.2 Inversion formulae......Page 39 3.3 The convolution theorem......Page 45 3.4 Limits of distribution functions......Page 51 3.5 The theorems of Helly......Page 53 3.6 The continuity theorem......Page 56 3.7 Infinite convolutions......Page 64 4.1 Necessary conditions......Page 77 4.2 Necessary and sufficient conditions......Page 79 4.3 Sufficient conditions......Page 92 4.4 Supplementary remarks concerning non-negative definite functions......Page 97 4.5 Unimodal distributions......Page 100 4.6 An essential property of characteristic functions......Page 108 5.1 Preliminary remarks on factorizations......Page 112 5.2 Definition of infinitely divisible characteristic functions......Page 116 5.3 Elementary properties of infinitely divisible characteristic functions......Page 117 5.4 Construction of infinitely divisible characteristic functions......Page 120 5.5 Canonical representations......Page 122 5.6 A limit theorem......Page 135 5.7 Characteristic functions of stable distributions......Page 137 5.8 Frequency functions of stable distributions......Page 147 5.9 Asymptotic expansions and integral representations of stable densities......Page 157 5.10 Unimodality of stable distributions......Page 167 5.11 Self-decomposnble distributions......Page 170 6.1 Some notations and lemmas......Page 175 6.2 General decomposition theorems......Page 178 6.3 Indecomposable characteristic functions......Page 189 7.1 The strip of regularity and the integral representation......Page 200 7.2 Analytic characteristic functions and their distribution functions......Page 206 7.3 Criteria for analytic characteristic functions......Page 219 7.4 Periodic analytic characteristic functions......Page 234 7.5 Analytic characteristic functions as solutions of certain differential equations......Page 236 8.1 Properties of the factors of an analytic characteristic function......Page 245 8.2 Factorization of certain entire characteristic functions......Page 252 8.3 Determination of certain entire characteristic functions by properties of their factors......Page 262 8.4 Infinitely divisible analytic characteristic functions......Page 267 9.1 The class L......Page 271 9.2 A sufficient condition for membership of I_0......Page 275 9.3 A necessary condition for membership of I_0......Page 289 9.4 Infinitely divisible characteristic functions with bounded Poisson spectrum......Page 290 9.5 Theorems concerning certain factorizations......Page 299 10.1 General theorems on \alpha-decompositions of analytic characteristic functions......Page 301 10.2 Special results concerning (X-decompositions......Page 310 11.1 The integral representation......Page 315 11.2 Infinitely divisible boundary characteristic functions......Page 319 12.1 Mixtures of distribution functions......Page 324 12.2 Transformations of characteristic functions......Page 327 Appendix A: The notations O and o......Page 336 Appendix C: Weierstrass' approximation theorem......Page 337 Appendix D: Order and type of entire functions......Page 338 Appendix E: Proof of lemmas needed in Chapter 9......Page 339 Appendix F: Schwarz's reflection principle......Page 343 List of Examples of Characteristic Functions......Page 344 REFERENCES......Page 346 INDEX......Page 353