arXiv subjects

Jianhang Ai

Publications and source records attributed to Jianhang Ai.

3 recordsLinked to original sources

arXiv2026

An entropic analogue of the MMS conjecture

Let $P=\{x_1,\ldots,x_n\}$ be a multiset consisting of $n\ge 2$ real numbers such that $\sum_{i=1}^{n}x_i=0$ and $\sum_{i=1}^{n}|x_i|>0$, and let $k <n$ be a positive integer. We sample $k$ elements from $P$ without replacement and set $X_P$ be the sum of the elements in our sample. It is shown that the Shannon entropy of $X_P$ satisfies \[ \mathbf{H}(X_P) \ge \mathbf{H}(\text{Ber}(k/n)) \, , \] where $\text{Ber}(k/n)$ is a Bernoulli random variable of mean $k/n$. The result is sharp, and may be seen as an entropic analogue of the Manickam-Miklós-Singhi (MMS) conjecture.

math.CO↗

arXiv2026

On lower bounds for hypergeometric tails

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 $1\le \mathbb{E}(H)\le \min\{i,k\}-1$ as well as that $\frac{(n-i)(n-k)}{n}>1$, it is shown that \[ \mathbb{P}(H\ge \mathbb{E}(H)) \,\ge\, \frac{e^{-1/12}}{4\sqrt{2}} \cdot \sqrt{\frac{n-1}{n}} \cdot\frac{ \sqrt{\text{Var}(H)} }{1 + \sqrt{1+ \frac{n-1}{n-k}\cdot\text{Var}(H)}}\, . \] 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.

math.PR↗

arXiv2025

Concentration inequalities for the sum in sampling without replacement: an approach via majorization

Let $P=(x_1,\ldots,x_n)$ be a population consisting of $n\ge 2$ real numbers whose sum is zero, and let $k <n$ be a positive integer. We sample $k$ elements from $P$ without replacement and denote by $X_P$ the sum of the elements in our sample. In this article, using ideas from the theory of majorization, we deduce non-asymptotic lower and upper bounds on the probability that $X_P$ exceeds its expected value.

math.PR↗