Wen-qing XU, Sha-sha WANG, Da-chuan XU
The classical Archimedean approximation of $\pi$ uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in $\mathbb{R}^2 $ and it is well-known that by using linear combinations of these basic estimates, modern extrapolation techniques can greatly speed up the approximation process. % reduce the associated approximation errors. Similarly, when $n$ vertices are randomly selected on the circle, the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge to $\pi$ almost surely as $ n \to \infty $, and by further applying extrapolation processes, faster convergence rates can also be achieved through similar linear combinations of the semiperimeter and area of these random polygons. In this paper, we further develop nonlinear extrapolation methods for approximating $\pi$ through certain nonlinear functions of the semiperimeter and area of such polygons. We focus on two types of extrapolation estimates of the forms $ \mathcal{X}_n = \mathcal{S}_n^{\alpha} \mathcal{A}_n^{\beta} $ and $ \mathcal{Y}_n (p) = \left( \alpha \mathcal{S}_n^p + \beta \mathcal{A}_n^p \right)^{1/p} $ where $ \alpha + \beta = 1 $, $ p \neq 0 $, and $ \mathcal{S}_n $ and $ \mathcal{A}_n $ respectively represents the semiperimeter and area of a random $n$-gon inscribed in the unit circle in $ \mathbb{R}^2 $, and $ \mathcal{X}_n $ may be viewed as the limit of $ \mathcal{Y}_n (p) $ when $ p \to 0 $. By deriving probabilistic asymptotic expansions with carefully controlled error estimates for $ \mathcal{X}_n $ and $ \mathcal{Y}_n (p) $, we show that the choice $ \alpha = 4/3 $, $ \beta = -1/3 $ minimizes the approximation error in both cases, and their distributions are also asymptotically normal.
