论文标题
方差膨胀和浆果 - 贝里（Berry-Esseen
Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon
论文作者
论文摘要
修复飞机上的容器多边形$ p $，并考虑凸船体$ p_n $ of $ n \ geq 3 $独立，并均匀分布在$ p $随机点中。本文的重点是随机多边形$ p_n $的顶点数。顶点数的精确方差扩展确定为恒定项，这一结果可以被视为Rényi和Sulanke（1963）期望的经典扩展的二阶类似物。此外，对于随机多边形$ p_n $的顶点编号，呈尖锐的浆果 - 埃森的界限，该数字与方差的正方形根部相同。两种结果证明背后的主要思想是将随机多边形$ p_n $的边界分解为随机凸形链，以及仔细合并方差扩展和浆果 - 埃塞恩的边界，以供各个链的顶点数字数。
Fix a container polygon $P$ in the plane and consider the convex hull $P_n$ of $n\geq 3$ independent and uniformly distributed in $P$ random points. In the focus of this paper is the vertex number of the random polygon $P_n$. The precise variance expansion for the vertex number is determined up to the constant-order term, a result which can be considered as a second-order analogue of the classical expansion for the expectation of Rényi and Sulanke (1963). Moreover, a sharp Berry-Esseen bound is derived for the vertex number of the random polygon $P_n$, which is of the same order as the square-root of the variance. The main idea behind the proof of both results is a decomposition of the boundary of the random polygon $P_n$ into random convex chains and a careful merging of the variance expansions and Berry-Esseen bounds for the vertex numbers of the individual chains.
