The convolution of two unimodal densities is not in general unimodal. In 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  had shown that the convolution of two symmetrical unimodal denstties is again symmetrical unimodal.Ibragimov  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.
KeywordsDensity Function Stochastic Process Convolution Private Communication Asymptotic Distribution
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