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

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.