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The convergence of distribution of integer-valued additive functions to the poisson law

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Abstract

Let {f x(m), x≥1} be a set of additive functions, and

$$v_x (fx(m))< u) = \frac{1}{{[x]}}\# \{ m \leqslant x,fx(m)< u\} .$$

Some results are obtained about the weak convergence of the distribution functionsv x(fx(m)<u) to the Poisson law.

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References

  1. G. Halasz, On the distribution of additive arithmetical functions,Acta Arithm.,27, 143–152 (1975).

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  2. B. V. Levin and N. M. Timofeev, On the distribution of values of additive functions,Acta Arithm.,26(4), 333–364 (1975).

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  3. J. Šiaulys, The von Mises theorem in number theory, in:New Trends in Probability and Statistics, Vol. 2: Analytic and Probabilistic Methods in Number Theory, F. Schweiger and E. Manstavičius (eds), VSP, Utrecht/TEV, Vilnius (1992). pp. 293–310.

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Authors
  1. J. Šiaulys
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Additional information

The research described in this paper was partially supported by a grant from the International Science Foundation.

Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 3, pp. 381–392, July–September, 1995.

Translated by J. Šiaulys

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Šiaulys, J. The convergence of distribution of integer-valued additive functions to the poisson law. Lith Math J 35, 300–308 (1995). https://doi.org/10.1007/BF02350365

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  • DOI: https://doi.org/10.1007/BF02350365

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