Central limit theorems for random polygons in an arbitrary convex set

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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 ⊂ R{double-struck}2. The novel aspect of our approach is that it yields uniform estimates for all convex sets K ⊂ R{double-struck}2 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.

Original languageEnglish (US)
Pages (from-to)881-903
Number of pages23
JournalAnnals of Probability
Volume39
Issue number3
DOIs
StatePublished - May 1 2011

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Central limit theorem
  • Random polygons

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