Convolution of Unimodal Distributions Can Produce any Number of Modes

Abstract

For any positive integer $n$, there exists a unimodal distribution $\mu$ such that $\mu \ast \mu$ is $n$-modal. Furthermore, there is a unimodal distribution $\mu$ such that $\mu \ast \mu$ has infinitely many modes. Lattice analogues of the results are also given.