A note on the maximum probability of ultra log-concave distributions
Heshan Aravinda
TL;DR
The paper investigates whether the maximum-probability inequality for ultra log-concave random variables, proved by Jakimiuk et al. for the case of integral mean, extends to $\mathbb{E}[X]>1$. It constructs an explicit ultra log-affine distribution with pmf $p(n)=C\frac{\lambda^n}{n!}$ on a finite interval (with $N\ge5$ and $\tfrac{3}{2}\le\lambda\le2$), yielding $1<\mathbb{E}[X]<2$ and $\max_n \mathbb{P}\{X=n\}=\mathbb{P}\{X=1\}$, while for $Z\sim \mathrm{Pois}(\mathbb{E}[X])$ the maximum occurs at $n=1$ with a strictly larger probability, so $\max_n \mathbb{P}\{X=n\} < \max_n \mathbb{P}\{Z=n\}$. A detailed two-point inequality, supported by bounds on partial sums and exponential terms (Facts I–II) and a monotonicity analysis of a function $h(\lambda,N)$, underpins the result for the stated parameter range. This demonstrates that the natural inequality can fail for $\mathbb{E}[X]>1$, clarifying the limits of max-probability comparisons within the ULC class. The discussion also notes related observations about related distributions and entropy-type bounds in the background literature.
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 $\mathbb{E}[X]>1$.