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On the structure of the Poisson trinomial distribution

Mark Broadie, Ina Petkova

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.

On the structure of the Poisson trinomial distribution

Abstract

We study sums of independent random variables that take values , , or . 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 of the unconditional mean. As a consequence, any two modes of the two conditional distributions are within of each other.
Paper Structure (7 sections, 15 theorems, 57 equations)

This paper contains 7 sections, 15 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Means
  3. Log-concavity and modes
  4. Generating functions and parity extraction
  5. Poisson binomial representation and mode--mean proximity
  6. The degenerate case
  7. Applications

Key Result

Theorem 1

Suppose $\mathbb P(X\in \mathbb Z)>0$ and $\mathbb P(X\in \mathbb Z+ \frac{1}{2})>0$. Then the distributions for $X\mid (X\in \mathbb Z)$ and $X\mid (X\in \mathbb Z + \frac{1}{2})$ are Poisson binomial, hence log-concave, with one or two modes. In particular, if $m_{\mathrm{even}}$ and $m_{\mathrm{o Moreover, and hence $|\mu^{\mathrm{even}} - \mu^{\mathrm{odd}}| \leq 1$. Similarly, and hence $|m

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Proposition \ref{['prop:bound']}
  • Lemma 3
  • proof
  • ...and 20 more