Random cyclic polygons from Dirichlet distributions and approximations of π

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.