On the Unimodality of High Convolutions of Discrete Distributions

It is shown that if $\{p_j\}$ is a discrete density function on the integers with support contained in $\{0, 1, \cdots, d\}$, 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 $\{p_j\}^{\ast_n}$ is unimodal for all $n \geq n_0$. Examples show that this result is nearly best possible, but weaker results are proved under less restrictive assumptions.