Limit theorems for skew products with mixing base maps and expanding on the average fibers without fiberwise centering

The paper’s Theorem 2.4 establishes both the existence/characterization of the asymptotic variance and a Berry–Esseen bound n^{-(1/(2+4γ))} under stretched-exponential α-mixing of the base, via multiple correlation estimates and the method of cumulants, together with the precise fiber BV Lasota–Yorke framework (Assumption 2.2) and a careful sub-σ-algebra transfer-operator argument for annealed dynamics. The candidate solution, while capturing much of the setup, makes critical mistakes: (i) it treats the base-only term φ̄(ω)=µ_ω(ϕ_ω) as if α-mixing applied directly to E[φ̄(ω)φ̄(σ^nω)], ignoring that µ_ω depends on the past coordinates so the relevant σ-algebras are not separated; the paper instead inserts transfer-operator approximations to produce future-measurable surrogates before invoking mixing (cf. its proof of Lemma 4.3 and the estimate (2.14)). (ii) It asserts a 1/√n Berry–Esseen bound for the martingale part without controlling the conditional variance term required by martingale BE theory. (iii) It also claims the annealed process is α-mixing with the same rate as the base, which the paper does not claim and is nontrivial. Hence the model’s proof is not correct at key steps, even though the end statements align with the paper’s main results.