On lower bounds for hypergeometric tails
Jianhang Ai, Christos Pelekis
TL;DR
The paper addresses lower bounds on the tail probability $\mathbb{P}(H \ge \mathbb{E}(H))$ for the hypergeometric variable $H \sim \text{Hyp}(n,i,k)$. It derives two principal results: a universal bound $\mathbb{P}(H \ge \mathbb{E}(H)) \ge \frac{k}{n}$ when $n \ge 8k$, and a variance-dependent bound involving $\mathrm{Var}(H)$ under mild conditions on the mean and the parameter regime, using a blend of binomial comparisons, Ehm's total-variation bound, and auxiliary results on MAD and tail conditional expectation. The work also introduces auxiliary results on mean absolute deviation and tail conditional expectation, which may be of independent interest and help bridge first-principles analysis with Berry-Esseen-type perspectives. It connects these bounds to the MMS conjecture and the Poisson-binomial representation of the hypergeometric distribution, highlighting the broader relevance to extremal tail behavior and potential improvements in related combinatorial inequalities.
Abstract
Let $n,k$ be positive integers such that $n\geq k$, and let $H$ be a hypergeometric random variable counting the number of black marbles in a sample without replacement of size $k$ from an urn that contains $i\in \{1,\ldots, n\}$ black and $n - i$ white marbles. It is shown that \[ \mathbb{P}(H \ge \mathbb{E}(H)) \ge k/n\, , \, \text{when} \,\, n\ge 8k \, . \] Furthermore, provided that $1\le \mathbb{E}(H)\le \min\{i,k\}-2$ as well as that $\frac{(n-i)(n-k)}{n}>1$, it is shown that \[ \mathbb{P}(H\ge \mathbb{E}(H)) \,\ge\, \frac{e^{-1/8}}{4\sqrt{2}} \cdot \sqrt{\frac{n-1}{n}} \cdot\frac{ \sqrt{\text{Var}(H)} }{1 + \sqrt{1+ \frac{n-1}{n-k}\cdot\text{Var}(H)}}\, . \] Auxiliary results which may be of independent interest include an upper bound on the tail conditional expectation and a lower bound on the mean absolute deviation of the hypergeometric distribution.