On the unimodality of high convolutions

Abstract. It has been conjectured, for any discrete density function (pj) on the integers, that there exists an no such that the n-fold con-volution (pj n is unimodal for all n> n. A similar conjecture has been stated for continuous densities. We present several counterexamples to both of these conjectures. As a positive result, it is shown for a discrete density with a conn cted 3-point integer support that its n-fold convolution is fully unimodal for all sufficiently large n