On the structure of the Poisson trinomial distribution

Abstract

We study sums of independent random variables that take values $0$, $1/2$, or $1$. We show that the probability mass function of the sum splits into two interleaved parts: one supported on the integers and the other supported on the half-integers. Each part, when normalized, is a Poisson binomial distribution and hence log-concave with one or two modes. We also prove that each of the two conditional means (conditioning on being an integer or a half-integer) lies within $1/2$ of the unconditional mean. As a consequence, any two modes of the two conditional distributions are within $5/2$ of each other.