Quenched invariance principle with a rate for random dynamical systems

The paper’s Theorem 3.2 states the self-normalized quenched invariance principle with Wasserstein rate W_{q/4}(W^ω_n, B) ≤ C n^{-1/4 + 1/(2q)} under (P1)–(P8) for a > 5 and q ≥ 4, with W^ω_n defined via the variance-profile time change N^ω_n(t) built from Σ_n^2(ω) as in equations (3.1)–(3.2) . The proof uses two martingale–coboundary decompositions: a primary decomposition ϕ = ψ + χ∘F − χ with ψ a reverse martingale difference (Lemma 4.1) and a secondary decomposition for ϕ̆ controlling sums of squares to compare the predictable quadratic variation with the deterministic profile, enabling the Skorokhod-embedding/time-change analysis (Sections 4–6) . By contrast, the model’s solution incorrectly asserts Σ_n^2(ω) = ∫(S_n^g)^2 dµ_ω + O(1) and uses E(V_k) = Σ_k^2(ω) for the martingale array built from g; the paper proves that Σ_n^2(ω)−η_n^2(ω) is in fact O_ω(n^{1/2} + n^{1/q}), not O(1) (Lemma 6.1) and carefully bridges the normalization/indexing mismatch via the secondary decomposition and maximal bounds . Without these ingredients, the model’s claimed n^{-1/2} coboundary reduction and its time-change control are not justified. The paper’s approach is internally consistent and aligns with the known n^{-1/4} barrier for Skorokhod embeddings (Remark 3.3) .