随机多边形质心的渐近正态性
Asymptotic Normality of Centroids of Random Polygons
AI总结：
本文研究由单位圆上顶点及随机排列长度构造的随机多边形质心的渐近行为，证明其经适当重标度后收敛于方差为1/12的圆对称复正态分布，体现了高度依赖随机变量和中的中心极限现象。
摘要：
我们研究了由单位圆上顶点构成的正多边形，通过延长射线使其长度形成前n个整数的随机排列而构造的随机多边形的质心的渐近行为。令人惊讶的是，这一问题与多种数学背景相关联，包括随机矩阵论和离散傅里叶变换。通过严谨的分析，我们证明了适当重标度后的质心序列收敛于方差为1/12的圆对称复正态分布。这一结果是中心极限行为在涉及高度依赖随机变量之和的情境中的体现。
英文摘要：
We explore the asymptotic behavior of the centroids of random polygons constructed from regular polygons with vertices on the unit circle by extending the rays so that their lengths form a random permutation of the first (n) integers. Surprisingly, this question has connections to diverse mathematical contexts, including random matrix theory and discrete Fourier transforms. Through rigorous analysis, we establish that the sequence of the suitably rescaled centroids converges to a circularly-symmetric complex normal distribution with variance (\frac{1}{12}). This result is a manifestation of central limit behavior in a setting involving sums of heavily dependent random variables.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
