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Metrika

, Volume 30, Issue 1, pp 165–170 | Cite as

Unimodality of differences

Article

Summary

The convolution of two unimodal densities is not in general unimodal. In [1953]Chung [see also his translation ofGnedenko/Kolmogorov] gave an example of i.i.d. random variablesX, Y, both with an unimodal densityf, whereX+Y has no unimodal density.Wintner [1938] had shown that the convolution of two symmetrical unimodal denstties is again symmetrical unimodal.Ibragimov [1956] proved the strong unimodality for the convolution of strongly unimodal densities.

For the differenceX-Y of two i.i.d. random variables with arbitrary densityf it is known and easily proved that it has a density which is symmetrical and maximal at 0. It seems to be not yet known and is proved in this paper that this density ofX-Y is unimodal iff is unimodal.

Keywords

Density Function Stochastic Process Convolution Private Communication Asymptotic Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Chung, K.L.: Sur les lois de probabilité unimodales. C.R. Acad. Sci. Paris236, 1953, 583–584.MathSciNetMATHGoogle Scholar
  2. Gnedenko, B.W., andA.N. Kolmogorov: Limit Distributions for sums of independent random variables; (translated, annotated and revised by Kai Lai Chung). Reading, Mass., 1954 (1968).Google Scholar
  3. Ibragimov, J.A.: On the composition of unimodal distributions (Russian), Teorija verojatnostej1, 1956, 283–288.MathSciNetMATHGoogle Scholar
  4. Wintner, A.: Asymptotic Distributions and Infinite Convolutions. Ann Arbor, Mich. 1938.Google Scholar

Copyright information

© Physica-Verlag 1983

Authors and Affiliations

  • H. Vogt
    • 1
  1. 1.Inst. f. Angew. Math. u. Statistik der Universität WürzburgWürzburg

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