BAYES POSTERIOR CONVERGENCE FOR LOSS FUNCTIONS VIA ALMOST ADDITIVE THERMODYNAMIC FORMALISM

The paper’s Theorem C proves (A) posterior convergence Π_n(·|y) → Π* by applying Kingman’s subadditive ergodic theorem to ψ_n(θ,·)±∫K_{θ,x}dµ_θ and then identifying ψ*(θ)=inf_n (1/n)∫ψ_n(θ,y)dν(y); dividing numerator and denominator by n yields the limit in the posterior ratio, as encoded in (42)–(44) and the proof text following (43) . For (B) the paper’s proof uses a ratio-control lemma (Lemma 4.2) together with a dynamical large-deviations theorem for almost-additive families (Theorem 3.6) to obtain the level-1 exponential bound and the strict negativity under the continuity condition, exactly as stated in (17) . The candidate solution establishes the same limit in (A) via Kingman (including an L^1 statement) and derives (B) by a different route: a block approximation plus a level-2 LDP for empirical measures (Kifer; Pfister–Sullivan). This approach is mathematically sound and leads to the same rate expression, but a few technical steps (notably a uniform-in-θ control in the block approximation and the switch from sup-norm to L^1 controls in the ratio) would need tightening. Overall, both reach the same conclusions; the paper’s argument is complete for the stated assumptions, while the model’s proof sketch is correct in substance but omits some technical justifications.