论文标题
kolmogorov的敏锐形式 - 罗戈因不平等和领导者的猜想 - 拉德克利夫
The sharp form of the Kolmogorov--Rogozin inequality and a conjecture of Leader--Radcliffe
论文作者
论文摘要
令$ x $为一个随机变量，并通过$$ \ mathcal {q} _ {h}（x）= \ sup_ {x \ in \ mathbb {r}} \ mathbb {p}（x \ in（x，x+h]）。变量Kolmogorov-Rogozin不平等指出$$ \ MATHCAL {q} _ {h}（s_n）\ leq c \ left（\ sum_ {i = 1}^{n}（1- \ mathcal {q} _ {h}（x_i））\ right）\ right）^{ - \ frac {1} {2} {2}}}} $$ 在本文中，我们给出了$ \ MATHCAL {q} _ {h}（s_n）$的最佳限制，以$ \ Mathcal {q} _ {h}（x_i）$来解决，该问题解决了一个由领导者和Radcliffe在1994年提出的问题。
Let $X$ be a random variable and define its concentration function by $$\mathcal{Q}_{h}(X)=\sup_{x\in \mathbb{R}}\mathbb{P}(X\in (x,x+h]).$$ For a sum $S_n=X_1+\cdots+X_n$ of independent real-valued random variables the Kolmogorov-Rogozin inequality states that $$\mathcal{Q}_{h}(S_n)\leq C\left(\sum_{i=1}^{n}(1-\mathcal{Q}_{h}(X_i))\right)^{-\frac{1}{2}}.$$ In this paper we give an optimal bound for $\mathcal{Q}_{h}(S_n)$ in terms of $\mathcal{Q}_{h}(X_i)$, which settles a question posed by Leader and Radcliffe in 1994. Moreover, we show that the extremal distributions are mixtures of two uniform distributions each lying on an arithmetic progression.
