A de Finetti-style Result for Polygons Drawn from the Symmetric Measure

TL;DR

This paper demonstrates that for large polygons, the distribution of small segments in open and closed polygonal chains drawn from the symmetric measure are very similar, with explicit bounds on their total variation.

Contribution

It provides a de Finetti-style result showing the similarity of segment distributions in open and closed polygons under the symmetric measure, with explicit quantitative bounds.

Findings

Small segments of open and closed polygons are statistically similar for large n.

Explicit bounds on total variation distance are established.

Results support the intuition of segment distribution similarity in polygon spaces.

Abstract

There is a natural intuition that, given a large $n$, the distributions of small segments of a randomly sampled polygonal chain and those of a randomly sampled closed polygonal chain (drawn from the subspace measure of course), should be very similar. We show that this is the case for the symmetric measure on polygon spaces, and provide explicit bounds on the total variation between these two distributions.