NON-UNIFORM BERRY-ESSEEN THEOREM AND EDGEWORTH EXPANSIONS WITH APPLICATIONS TO TRANSPORT DISTANCES FOR WEAKLY DEPENDENT RANDOM VARIABLES

The paper states and proves a non-uniform Berry–Esseen bound under Assumption 3 and non-uniform Edgeworth expansions under Assumptions 3 and 6, with the precise dependence of the Edgeworth polynomials on An, Bn−σn, and Λn^{(j)}(0)σn^{j−2} (Theorems 5 and 8). Its proof uses a smoothing inequality (Eq. (9.6)) and Fourier-analytic estimates (Propositions 60–61) to obtain |Fn(x)−Φ(x)| ≤ Cσn^{-1}(1+|x|)^{-m}, and, for r≤m−2, |Fn(x)−Ψr,n(x)| ≤ Cnσn^{-r}(1+|x|)^{-m} with Cn→0, plus the Lp consequences and the explicit Hermite-based form in the self-normalized case . The candidate solution follows the same Fourier scheme: (i) local control of the characteristic function and its derivatives from Assumption 3 (matching Lemma 64) , (ii) an m-fold integration-by-parts smoothing inequality yielding the (1+|x|)^{-m} factor (the paper uses a closely related smoothing bound (9.6)) , (iii) Taylor expansion of log f_n to build Edgeworth polynomials with coefficients depending on standardized cumulants and the shift/scale parameters (matching Section 9.1 and Eq. (2.3)) . One minor discrepancy is the model’s shorthand treatment of the smoothing remainder: it uses a cutoff-based integration-by-parts variant giving a T^{-m} truncation term, whereas the paper’s Lemma (9.6) features a K/T term and therefore chooses T≈σn^{m−2} in the expansion proof (to force K/T=o(σn^{-r})) . Aside from this technical presentation difference (easily reconciled), the arguments and results agree.