Nonlinear random extrapolation estimates of \(\pi\) 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) \( S_n' \) of the corresponding random circumscribing polygons are known to converge to \( \pi \) w.p.\(1\) and their distributions are also asymptotically normal as \( n \to \infty \), we study in this paper nonlinear extrapolations of the forms \( \mathcal{W}_n = S_n^{\alpha} A_n^{\beta} S_n'^{\, \gamma} \) and \( \mathcal{W}_n (p) = ( \alpha S_n^p + \beta A_n^p + \gamma S_n'^{\, p} )^{1/p} \) where \( \alpha + \beta + \gamma = 1 \) and \( p \neq 0 \). 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.

Keywords

• Random polygons
• Nonlinear extrapolations
• Dirichlet distribution
• Central limit theorems
• Cram\'{e}r's theorem