Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon

TL;DR

This paper provides a detailed variance expansion and a sharp Berry-Esseen bound for the number of vertices of a random polygon formed by points uniformly distributed within a fixed polygon, advancing understanding of geometric probability.

Contribution

It offers the first second-order variance expansion and a precise Berry-Esseen bound for the vertex count of random polygons in a fixed polygon.

Findings

Variance expansion up to constant order for vertex number

Sharp Berry-Esseen bound proportional to the square root of variance

Decomposition method using convex chains for analysis

Abstract

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 R\'enyi 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.