Random cyclic polygons from Dirichlet distributions and approximations of π

Shasha Wang and Wen-Qing Xu

Statistics & Probability Letters, 2018, vol. 140, issue C, 84-90

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.

Keywords: Dirichlet distributions; Random cyclic polygons; Approximations of π; Central limit theorems; Extrapolation (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1016/j.spl.2018.05.007

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