Mixing rates for potentials of non-summable variations

The paper’s Theorem 1 constructs a blockwise maximal coupling and proves P[X_n=1] ≤ C n^{-(βδ+1)/2} by: (i) a KL chain-rule estimate of the block divergence via the χ^2-variation rates (Lemma 1), (ii) TV bounds (Pinsker/Bretagnolle–Huber), (iii) a renewal/hazard process built from q_{n,k} and its envelope b_k, and (iv) a renewal estimate that yields the stated rate . The candidate’s proof sketches a different route: it tries to bound the block χ^2-divergence multiplicatively and then deduce TV ≤ (1/2)√χ^2. However, its key tensorization step mis-indexes the χ^2-variation rate (using absolute time t instead of the number of matched coordinates t−M_n), which incorrectly makes the tail sum decay with n even without incorporating the renewal offset k. This bypasses the core renewal/hazard machinery essential in the paper and leads to an unjustified n-dependent decay. Hence, while the final rate matches Theorem 1, the argument as written is not valid; the paper’s argument is correct and complete, while the model’s is flawed.