Central limit theorems for random polygons in an arbitrary convex set

Details Central limit theorems for random polygons in an arbitrary convex set Central limit theorems for random polygons in an arbitrary convex set

2010-03-22

arxiv.org/abs/1003.4209v3

Annals of Probability 2011, Vol. 39, No. 3, 881-903

Mathematics

Probability

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

Scientific paper

10.1214/10-AOP568

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.

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