Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon
Annales Henri Lebesgue, Volume 6 (2023), pp. 875-906.

Keywords Berry–Esseen bound, central limit theorem, geometric probability, Poisson point process, random convex chain, random polygon, variance expansion

### Abstract

Fix a container polygon $P$ in the plane and consider the convex hull ${P}_{n}$ of $n\ge 3$ 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 one over the square-root of the variance. The latter is optimal and improves the earlier result of Bárány and Reitzner (2006) by removing the factor ${\left(loglogn\right)}^{60}$ in the planar case. 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. In the course of the proof, we derive similar results for the Poissonized model.

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