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On the limit distribution of discrete random variables
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  • Published: May 1987

On the limit distribution of discrete random variables

  • Adolf Hildebrand1 

Probability Theory and Related Fields volume 75, pages 67–76 (1987)Cite this article

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Summary

It is well-known that the possible limit distributions of triangular arrays (X nk )(n≧1, 1≦k≦k n )of infinitesimal random variables with independent rows are exactly the infinitely divisible distributions. In the present paper, it is shown that requiring the random variables X nk to assume only two values with prescribed probabilities p nk and 1-p nk does not narrow the class of possible limit distributions, provided that the numbers p nk satisfy 0<p nk <1 and

$$\mathop {lim}\limits_{n \to \infty } \sum\limits_{\mathop {1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } k\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } kn}\limits_{Pnk\underline \leqslant \varepsilon } } {P_{nk} = \infty } for every \varepsilon > 0.$$

A similar result is obtained for limit distributions of sequences of inde-pendent random variables. As an application to probabilistic number theory, the class of possible limit distributions in the central limit theorem for additive arithmetic functions is determined.

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References

  1. Elliott, P.D.T.A.: Probabilistic number theory I, II. Berlin, Heidelberg, New York: Springer 1979, 1980

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  2. Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Reading Mass.: Addison-Wesley 1968

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  3. Lukacs, E.: Characteristic functions. London: Griffin 1970

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  4. Urbanik, K.: Problem 272. Colloq. Math. 6, 336 (1958)

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 61801, Urbana, USA

    Adolf Hildebrand

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  1. Adolf Hildebrand
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Hildebrand, A. On the limit distribution of discrete random variables. Probab. Th. Rel. Fields 75, 67–76 (1987). https://doi.org/10.1007/BF00320081

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  • Revised: 01 September 1986

  • Issue Date: May 1987

  • DOI: https://doi.org/10.1007/BF00320081

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Keywords

  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Number Theory
  • Statistical Theory
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