Random Polygons From Dirichlet Distributions And Extrapolation Estimations Of π

The classical Archimedean approximation ofπuses the semiperimeter of regular poly-gons inscribed in or circumscribed about a unit circle in R~2.With the doubling of the number of sides of these regular polygons,Archimedes discovered his famous harmonic-geometric-mean relations satisfied by the semiperimeter of the regular n-sided polygons and regular 2n-sided polygons.Through these recurrence relations,he calculated these quantities for n=6,12,24,48,96 and obtained the famous estimates 223/71<π<22/7.Similarly,when the vertices are randomly selected on the circle,the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are expected to converge toπwith probability1 as n→∞,and their distributions asymptotically Gaussian.This has been formally verified for the cases when the vertices are independently and uniformly distributed or follow the multi-variate symmetric Dirichlet distributions on the circle.Additionally,by applying extrapolation estimates,faster rates of convergence can also be obtained through the linear combinations of the semiperimeter and area of these random polygons.In this paper,we consider approximations ofπbased on random polygons from symmetric Dirichlet distributions.By using extrapolation methods,we construct linear combinations of the semiperimeter and area of the inscribed random polygons on a unit circle to improve the approximations ofπ.However the process of doubling of the sides of the random polygons is more complicated and different doubling process can lead to different convergence results.We consider three different doubling methods,namely,independent doubling,equal bisection and random bisection of the random polygons.By suitably doubling the number of vertices of the underlying random n-gons and constructing optimal linear combinations of the semiperimeter and area of these random polygons,we derive several extrapolation improvements with much faster rates of convergence.The semiperimeter and area of these random n-gons and 2n-gons and the resulting extrapolation improvements all converge toπwith probability 1 as n→∞,and their distributions are also asymptotically Gaussian.