## Infinite Divisibility of Probability Distributions on the Real LineInfinite Divisibility of Probability Distributions on the Real Line reassesses classical theory and presents new developments, while focusing on divisibility with respect to convolution or addition of independent random variables. This definitive, example-rich text supplies approximately 100 examples to correspond with all major chapter topics and reviews infinite divisibility in light of the central limit problem. It contrasts infinite divisibility with finite divisibility, discusses the preservation of infinite divisibility under mixing for many classes of distributions, and investigates self-decomposability and stability on the nonnegative reals, nonnegative integers, and the reals. |

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A real collection of useful results.

See p. 139, Proposition 2.4.

### Contents

1 | |

CHAPTER II INFINITELY DIVISIBLE DISTRIBUTIONS ON THE NONNEGATIVE INTEGERS | 23 |

CHAPTER III INFINITELY DIVISIBLE DISTRIBUTIONS ON THE NONNEGATIVE REALS | 77 |

CHAPTER IV INFINITELY DIVISIBLE DISTRIBUTIONS ON THE REAL LINE | 135 |

CHAPTER V SELFDECOMPOSABILITY AND STABILITY | 221 |

CHAPTER VI INFINITE DIVISIBILITY AND MIXTURES | 327 |

CHAPTER VII INFINITE DIVISIBILITY IN STOCHASTIC PROCESSES | 425 |

APPENDIX A PREREQUISITES FROM PROBABILITY AND ANALYSIS | 465 |

APPENDIX B SELECTED WELLKNOWN DISTRIBUTIONS | 503 |

513 | |

535 | |

539 | |

NOTATIONS AND CONVENTIONS | 545 |

### Other editions - View all

Infinite Divisibility of Probability Distributions on the Real Line Fred W. Steutel,Klaas van Harn No preview available - 2003 |

### Common terms and phrases

22 Example 270 Madison Avenue absolutely continuous absolutely monotone canonical density canonical function canonical representation canonical sequence canonical triple compound-exponential distributions compound-geometric compound-Poisson consider converse Corollary deﬁned degenerate distribution distrib distribution on Z+ distribution pk divisible characteristic function divisible distribution function equation exponential distributions ﬁnite ﬁrst func function F gamma convolution gamma distributions gamma(r geometric distributions hence inﬁnitely divisible hyperbolically completely monotone iﬀ independent inequality inﬁn infinitely inﬁnitely divisible characteristic inﬁnitely divisible distribution inﬁnitely divisible pgf inﬁnitely divisible pLSt k∈Z+ L´evy Laplace distributions Lemma Let F log-concave log-convex Marcel Dekker mixture of exponential monotone functions nondecreasing nonincreasing nonnegative numbers obtained parameter pgf’s pLSt’s Poisson distributions Proposition proved R+-valued random variable random variable satisﬁes Section self-decomposable distribution semigroup sii-process stable distributions stable with exponent tail unimodal ution York Z+-valued zero