Central limit theorems for random polygons in an arbitrary convex set

Abstract

We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset {\mathbb{R}}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets $K\subset {\mathbb{R}}^2$ without imposing any regularity conditions on the boundary $\partial K$. Our main result is a central limit theorem for both the area and the number of vertices, settling a well-known conjecture in the field. We also obtain asymptotic results relating the growth of the expectation and variance of these two functionals.

Published in at http://dx.doi.org/10.1214/10-AOP568 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Poisson polytopes
• Central limit theorems for uniform model random polygons

Cited by in corpus (3)

• Brownian limits, local limits and variance asymptotics for convex hulls in the ball
• Interaction of Poisson hyperplane processes and convex bodies
• On random convex chains, orthogonal polynomials, PF sequences and probabilistic limit theorems