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

Unimodality of differences



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.


Density Function Stochastic Process Convolution Private Communication Asymptotic Distribution 


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  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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