Estimation of Stationary Optimal Transport Plans

The paper proves that there exists a block-length schedule k(n) with k(n)→∞ such that the empirical per-letter k-block optimal transport cost converges almost surely to the optimal stationary joining value S(c;µ,ν), and the empirical block joinings converge weakly to the set of optimal joinings Jmin(µ,ν) (Theorem 3.1), using the notions of c-admissibility and a key reduction to the block OT problem together with Proposition 2.2 (lim_k k^{-1}T(ck;µ_k,ν_k)=S) . The candidate’s solution establishes the same claims via a different route: (i) a direct compactness/seam-bound argument to prove lim_k ρ_k=S, (ii) continuity of T(ck;·,·) in total variation for fixed k, and (iii) a deterministic Egorov/Borel–Cantelli diagonalization to choose k(n), plus an explicit seam-bound/Lipschitz control to identify subsequential limits as optimal joinings. Both approaches are correct; the model’s proof is self-contained and technically sound, while the paper’s proof is framed around the adapted costs and admissibility machinery and cites auxiliary propositions to streamline the argument .