A note on the maximum probability of ultra log-concave distributions

TL;DR

This paper investigates the maximum probability bounds of ultra log-concave distributions, showing that a previously established inequality does not hold universally when the mean exceeds one.

Contribution

It demonstrates the limitations of a known inequality for ultra log-concave distributions with mean greater than one, refining understanding of their probability bounds.

Findings

The inequality holds for ultra log-concave variables with integral mean.

Counterexamples show the inequality fails when the mean exceeds one.

The result clarifies the scope of probability bounds for ultra log-concave distributions.

Abstract

Jakimiuk et al. (2024) have proved that, if $X$ is an ultra log-concave random variable with integral mean, then $$\max_n \mathbb{P}\{X=n\} \geq \max_n \mathbb{P} \{Z=n\}\,,$$ where $Z$ is a Poisson random variable with the parameter $\mathbb{E}[X]$. In this note, we show that this inequality does not always hold true when $X$ is ultra log-concave with $\mathbb{E}[X]>1$.