Central limit theorems for uniform model random polygons

TL;DR

This paper demonstrates that a central limit theorem for Poisson model random polygons can be extended to uniform model polygons by showing their expectations and variances are asymptotically equivalent, using integral geometric methods.

Contribution

It establishes a link between CLTs for Poisson and uniform models of random polygons through asymptotic expectation and variance analysis.

Findings

CLT for Poisson model implies CLT for uniform model

Asymptotic equivalence of expectations and variances proven

Integral geometric expressions used for estimates

Abstract

We show how a central limit theorem for Poisson model random polygons implies a central limit theorem for uniform model random polygons. To prove this implication, it suffices to show that in the two models, the variables in question have asymptotically the same expectation and variance. We use integral geometric expressions for these expectations and variances to reduce the desired estimates to the convergence $(1+\frac\alpha n)^n\to e^\alpha$ as $n\to\infty$.