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It is shown that if { p j } is a discrete density function on the integers with support contained in {0 , 1 , ... , d }, and p 0 > 0, p 1 > 0, p d - 1 > 0, p d > 0, then there is an n 0 such that the n-fold convolution { p j }*n is unimodal for all n ♢ n 0. Examples show that this result is nearly best possible, but weaker results are proved under less restrictive assumptions.

On the Unimodality of High Convolutions of Discrete Distributions