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Random cyclic polygons from Dirichlet distributions and approximations of π

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  • Wang, Shasha
  • Xu, Wen-Qing

Abstract

The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in R2. When n vertices are independently and uniformly randomly selected on the circle, a random inscribed or circumscribing polygon can be constructed and it is known that their semiperimeter and area both converge to π almost surely as n→∞ and their distributions are also asymptotically Gaussian. In this paper, we extend these results to the case of random cyclic polygons generated from symmetric Dirichlet distributions and show that as n→∞, similar convergence results hold for the semiperimeters or areas of these random polygons. Additionally, we also present some extrapolation estimates with faster rates of convergence.

Suggested Citation

  • Wang, Shasha & Xu, Wen-Qing, 2018. "Random cyclic polygons from Dirichlet distributions and approximations of π," Statistics & Probability Letters, Elsevier, vol. 140(C), pages 84-90.
    • Handle: RePEc:eee:stapro:v:140:y:2018:i:c:p:84-90
    DOI: 10.1016/j.spl.2018.05.007
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