## Characteristic FunctionsPreliminary study of characteristic functions; Fundamental properties of characteristic functions; Criteria for characteristic functions; Factorization problems - infinitely divisible characteristic functions; Factorization problems - general theorems from the arithmetic of distribution functions; Analytic characteristic functions; Factorization of analytic characteristic functions; Infinitely divisible characteristic functions without indecomposable factors; Decompositions; Boundary characteristic functions; Mixtures of distribution functions and transformation of characteristic functions. |

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### Contents

INTRODUCTION | 1 |

PRELIMINARY STUDY OF CHARACTERISTIC FUNCTIONS | 15 |

FUNDAMENTAL PROPERTIES OF CHARACTERISTIC FUNCTIONS | 28 |

Copyright | |

12 other sections not shown

### Common terms and phrases

absolutely continuous absolutely continuous distribution analytic characteristic function assume assumption boundary characteristic function bounded variation Cauchy distribution charac characteristic function f(t coefficients conclude continuity theorem convergence Corollary to theorem decomposition defined degenerate distribution denote derive discontinuity points distribution function F(x divisible characteristic function easily seen entire characteristic function entire function equation example exists Fn(x formula frequency function function of order function without zeros Gamma distribution half-plane indecomposable distribution indecomposable factors indirect proof inequality infinite convolution infinitely divisible characteristic infinitely divisible distribution integer lattice distribution lemma Let F Let F(x moments Moreover non-decreasing function normal distribution parameters Poisson distribution Poisson spectrum positive constant positive integer positive number possible proof of theorem prove purely discrete distribution radius of convergence saltus satisfies the conditions singular distribution stable distributions statement strip of regularity sufficient condition sufficiently large symmetric teristic function tion unimodal distribution valid write