A note on the binomial distribution motivated by Chvátal's theorem and Tomasewski's theorem
Zheng-Yan Guo, Ze-Chun Hu, Run-Yu Wang
TL;DR
Problem: characterize the least concentration probability $f_n(k)=P(|B(n,k/n)-k|\le sqrt(Var(B(n,k/n))))$ over $k=0,...,n$ for fixed $n$. Approach: combine symmetry, Berry-Esseen-based normal approximation, CLT-type bounds, and exhaustive finite-n checks. Main result: the minimum occurs at $k=1$ and $k=n-1$ with $f_n(1)=f_n(n-1)=1$ if $n=1$ and $((n-1)/n)^{n-1}$ for $n\ge 2$. Significance: connects to Chvátal's and Tomaszewski's theorems and provides precise concentration bounds for the binomial distribution, including a practical explicit form for the minimum and a detailed finite-n analysis.
Abstract
Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chvátal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,1,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $2n/3$. Let $\mathcal{R}$ be the family of random variables of the form $X=\sum^n_{k=1}a_k\varepsilon_k$, where $n\ge 1$, $a_k, k=1, \dots, n,$ are real numbers with $\sum^n_{k=1} a_k^2=1$, and $\varepsilon_k$, $k=1, 2, \dots$, are independent Rademacher random variables (i.e., $P(\varepsilon_k=1)=P(\varepsilon_k=-1)=1/2$). Tomaszewski's theorem says that $\inf_{X\in \mathcal{R}}P(|X|\leq 1)=1/2$. Motivated by Chvátal's Theorem and Tomasewski's Theorem, in this note, we study the minimum value of the probability $f_n(k):=P(|B(n,k/n)-k|\leq \sqrt{{\rm Var} (B(n,k/n))})$ when $k$ ranges over $\{0,1,\ldots,n\}$ for any fixed $n\geq 1$, where ${\rm Var} (\cdot)$ denotes the variance, and prove that it is the smallest when $k=1$ and $n-1$.