Is the sum of independent unimodal random variables still unimodal? If yes, can you please give me some hint on why this holds? If no, can you show me some counter-example and suggest under what condition the sum remains unimodal? Thank you in advance.
Is the sum of independent unimodal random variables still unimodal?
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2The key concept (and term) here is *strong unimodality*. For continuous distributions, see I. A. Ibragimov (1956), [On the composition of unimodal distributions](http://link.aip.org/link/?TPRBAU/1/255/1), *Theory Prob. Appl.*, vol. 1, pp. 255-260, and for the discrete counterpart, J. Keilson and H. Gerber (1971), [Some results for discrete unimodality](http://www.jstor.org/pss/2283941), *JASA*, vol. 66, no. 334, pp. 386-389. – 2011-10-11