论文标题
关于离散log-concave发行的Feige的猜想
On a Conjecture of Feige for Discrete Log-Concave Distributions
论文作者
论文摘要
Feige（2006）的一个显着猜想断言，对于任何$ n $独立的非负随机变量$ x_1，x_2，\ dots，x_n $，每个变量最多都有期望，最多有$ 1 $，$ 1 $ \ mathbb {p} = \ sum_ {i = 1}^n x_i $。在本文中，我们调查了该类别的离散对数符合概率分布类别的猜想，并证明了加强版本。更具体地说，我们表明，当$ x_i $的$ x_i $是独立的离散log-concave时，猜想的$ 1/e $持有，并具有任意期望。
A remarkable conjecture of Feige (2006) asserts that for any collection of $n$ independent non-negative random variables $X_1, X_2, \dots, X_n$, each with expectation at most $1$, $$ \mathbb{P}(X < \mathbb{E}[X] + 1) \geq \frac{1}{e}, $$ where $X = \sum_{i=1}^n X_i$. In this paper, we investigate this conjecture for the class of discrete log-concave probability distributions and we prove a strengthened version. More specifically, we show that the conjectured bound $1/e$ holds when $X_i$'s are independent discrete log-concave with arbitrary expectation.
