Asymptotic Normality of Centroids of Random Polygons

TL;DR

This paper proves that the rescaled centroids of certain random polygons converge to a circularly-symmetric complex normal distribution, revealing a central limit phenomenon in a dependent random variable setting.

Contribution

It establishes the asymptotic normality of centroids of random polygons with vertices on the unit circle, connecting geometric probability with central limit theory.

Findings

Rescaled centroids converge to a complex normal distribution

Variance of the limit distribution is 1/12

Results link random polygon geometry to random matrix theory

Abstract

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.