On the structure of the Poisson trinomial distribution

TL;DR

This paper analyzes the structure of the Poisson trinomial distribution, revealing its decomposition into two interleaved Poisson binomial distributions supported on integers and half-integers, with implications for modes and means.

Contribution

It characterizes the Poisson trinomial distribution's structure, showing its probability mass splits into two log-concave parts with specific mean and mode properties.

Findings

Distribution splits into integer and half-integer supported parts

Each part is a Poisson binomial distribution with log-concavity

Conditional means are within 1/2 of the unconditional mean

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.