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Infinitely Divisible Laws

  • Yuan Shih Chow
  • Henry Teicher
Chapter
  • 506 Downloads
Part of the Springer Texts in Statistics book series (STS)

Abstract

It is a remarkable fact that the class of limit distributions of normed sums of i.i.d. random variables is severely circumscribed. If the underlying r.v.s, say {X n , n ≥ 1} have merely absolute moments of order r, then for r ≥ 2 only the normal distribution can arise as a limit, while if 0 < r ≤ 2, the limit law belongs to a class called stable distributions. If the basic r.v.s are merely independent (and infinitesimal when normed cf. (1) of Section 2), a larger class of limit laws, the so-called class ℒ emerges. But even the class ℒ does not contain a distribution of such crucial importance as the Poisson. A perusal of the derivation (Chapter 2) of the Poisson law as a limit of binomial laws B n reveals that the success probability associated with B n is a function of n. Thus, if B n-1 is envisaged as the distribution of the sum of i.i.d. random variables Y l, …, Y n-1, then B n must be the distribution of the sum of n different i.i.d. random variables which, therefore, may as well be labeled X n, 1, X n, 2,..., X n, n .

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References

  1. K. L. Chung, A Course in Probability Theory, Harcourt Brace, New York, 1968; 2nd ed., Academic Press, New York, 1974.zbMATHGoogle Scholar
  2. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, Mass., 1954.zbMATHGoogle Scholar
  3. P. Lévy, Théorie de l’addition des variables aléatoires Garthier-Villars, Paris, 1937; 2nd ed., 1954.Google Scholar
  4. M. Loève, Probability Theory, 3rd ed., Van Nostrand, Princeton, 1963; 4th ed., Springer-Verlag, Berlin and New York, 1977–1978.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Yuan Shih Chow
    • 1
  • Henry Teicher
    • 2
  1. 1.Department of Mathematical StatisticsColumbia UniversityNew YorkUSA
  2. 2.Department of StatisticsRutgers UniversityNew BrunswickUSA

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