# Nonlinear random extrapolation estimates of $$\pi$$ under Dirichlet distributions

## Authors

• Shasha Wang Department of Mathematics and Physics, Shijiazhuang Tiedao University, China https://orcid.org/0009-0009-1303-5871
• Zecheng Li Questrom School of Business, Boston University, USA https://orcid.org/0009-0007-7899-1704
• Wen-Qing Xu Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, China & Department of Mathematics and Statistics, California State University Long Beach, USA https://orcid.org/0000-0002-0080-002X

## Keywords:

Random polygons, Nonlinear extrapolations, Dirichlet distribution, Central limit theorems, Cram\'{e}r's theorem
Abstract views: 61

## 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.

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W.-Q. Xu, S. Wang, D. Xu, Nonlinear extrapolation estimates of π, Acta Math. Sin. Appl. Engl. Ser., to appear.

2023-12-28

## How to Cite

Wang, S., Li, Z., & Xu, W.-Q. (2023). Nonlinear random extrapolation estimates of $$\pi$$ under Dirichlet distributions. J. Numer. Anal. Approx. Theory, 52(2), 273–294. https://doi.org/10.33993/jnaat522-1360

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