The Annals of Probability

Central limit theorems for random polygons in an arbitrary convex set

John Pardon

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Abstract

We study the probability distribution of the area and the number of vertices of random polygons in a convex set K⊂ℝ2. The novel aspect of our approach is that it yields uniform estimates for all convex sets K⊂ℝ2 without imposing any regularity conditions on the boundary ∂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.

Article information

Source
Ann. Probab., Volume 39, Number 3 (2011), 881-903.

Dates
First available in Project Euclid: 16 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1300281727

Digital Object Identifier
doi:10.1214/10-AOP568

Mathematical Reviews number (MathSciNet)
MR2789578

Zentralblatt MATH identifier
1221.52011

Subjects
Primary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems

Keywords
Random polygons central limit theorem

Citation

Pardon, John. Central limit theorems for random polygons in an arbitrary convex set. Ann. Probab. 39 (2011), no. 3, 881--903. doi:10.1214/10-AOP568. https://projecteuclid.org/euclid.aop/1300281727


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