Proof of the Diaconis--Freedman Conjecture on partially-exchangeable processes

We prove a conjecture of Diaconis and Freedman (Ann. Probab. 1980) characterising the extreme points of the set of partially-exchangeable processes on a countable set. More concretely, we prove that the partially exchangeable sigma-algebra of any transient partially exchangeable process $X=(X_i{)}_{i\ge 0}$ (and hence any transient Markov chain) coincides up to null sets with the sigma-algebra generated by the initial state $X_0$ and the transition counts $(\mathrm{\#}\{i\ge 0:X_i=x,X_{i+1}=y\}:x,y\in S)$. Our proof is based on an analysis of Gibbs measures for Eulerian paths on rooted digraphs, relying in particular on the connection to uniform spanning trees and Wilson’s algorithm via the de Bruijn–Ehrenfest–Smith–Tutte (BEST) bijection, and yields an explicit method to sample from the conditional distribution of a transient Markov chain given its transition counts.