Nonlinear random extrapolation estimates of π under Dirichlet distributions

Abstract

We construct optimal nonlinear extrapolation estimates of $\pi$ based on random cyclic polygons generated from symmetric Dirichlet distributions. While the semiperimeter $S_n$ and the area $A_n$ of such random inscribed polygons and the semiperimeter (and area) $S_n^{\prime }$ of the corresponding random circumscribing polygons are known to converge to $\pi$ w.p.$1$ and their distributions are also asymptotically normal as $n\to \mathrm{\infty }$, we study in this paper nonlinear extrapolations of the forms ${\mathcal{W}}_n=S_n^{\alpha }{A}_n^{\beta }{S}_n^{\prime \gamma }$ and ${\mathcal{W}}_n(p)=(\alpha S_n^p+\beta A_n^p+\gamma S_n^{\prime p}{)}^{1/p}$ where $\alpha +\beta +\gamma =1$ and $p\ne 0$. By deriving probabilistic asymptotic expansions with carefully controlled error estimates, we show that ${\mathcal{W}}_n$ and ${\mathcal{W}}_n(p)$ also converge to $\pi$ w.p.$1$ and are asymptotically normal. Furthermore, to minimize the approximation error associated with ${\mathcal{W}}_n$ and ${\mathcal{W}}_n(p)$, the parameters must satisfy the optimality condition $\alpha +4\beta -2\gamma =0$. Our results generalize previous work on nonlinear extrapolations of $\pi$ which employ inscribed polygons only and the vertices are also assumed to be independently and uniformly distributed on the unit circle.