Random Polygons and Optimal Extrapolation Estimates of pi

We construct optimal extrapolation estimates of π based on random polygons generated by n independent points uniformly distributed on a unit circle in R2. While the semiperimeters and areas of these random n-gons converge to π almost surely and are asymptotically normal as n → ∞, in this paper we develop various extrapolation processes to further accelerate such convergence. By simultaneously considering the random n-gons and suitably constructed random 2n-gons and then optimizing over functionals of the semiperimeters and areas of these random polygons, we derive several new estimates of π with faster convergence rates. These extrapolation improvements are also shown to be asymptotically normal as n → ∞.