超越两倍均值的尖锐尾界

中文摘要 AI 辅助

考虑n个独立、非负且均值至多为1的随机变量X₁,X₂,…，本文证明其总和超过阈值t的概率满足：对所有t≥2n+1，有P[∑ᵢ₌₁ⁿXᵢ≥t]≤1−(1−1/t)ⁿ。为证明该结论，我们对一组有序但非独立随机变量的序列求解松弛优化问题，将其递归重构为动态规划问题。当二元独立同分布随机变量满足P[Xᵢ=0]=1−1/t、P[Xᵢ=t]=1/t时，该界取等，且此分布仍是松弛问题中的最大化者。

英文摘要

Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \[ \mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,. \] To prove this, we consider a relaxed optimization problem over a set of sequences of ordered, but non-independent random variables. This allows us to reformulate it recursively as dynamic programming problem. The bound becomes an equality for the binary i.i.d.~random variables satisfying $\mathbb{P}\left[X_i=0\right]= 1-\frac{1}{t}$ and $\mathbb{P}\left[X_i=t\right]=\frac{1}{t}$, which remains the maximizer in the relaxed problem.

发表机构

• Yale University（耶鲁大学）

机构由 AI 辅助整理，请以论文原文为准。