Berry-Esseen bounds with targets for products of random matrices

The paper states and proves optimal O(n^{-1/2}) Berry–Esseen bounds with targets for (σ(S_n,x), S_nx) and for matrix coefficients (log|⟨S_n v,f⟩|, S_nx) under the standard proximal/strongly-irreducible and finite exponential moment hypotheses (Theorems 1.1–1.2) . The argument rests on (i) Le Page’s spectral perturbation theory for the Fourier-perturbed transfer operators on C^α(P^{d−1}) with the spectral decomposition P_z = λ_z N_z + Q_z and precise expansions for λ_z (Proposition 2.4; Lemma 2.5) , and (ii) a key smoothing step via a Cauchy principal value identity (Lemma 2.8 as used in (4.10)) to remove the ξ=0 singularity without resorting to complex contours , combined with a careful frequency localization and normalization (dn,x) in the ψ=1 case (Section 3; Proposition 3.2) . For coefficients, the proof uses the exact decomposition log(|⟨S_n v,f⟩|/(||v|| ||f||)) = σ(S_n,x) + log d(S_nx, H_y) (eq. (4.2)) and a partition of unity near the hyperplane H_y (Lemma 4.1), together with non-concentration of ν near hyperplanes (Proposition 2.2) and large-deviation inputs (Proposition 2.1), to control the singular region and still achieve O(n^{-1/2}) uniformly (Proposition 4.2 and Lemmas 4.7–4.10) . By contrast, the candidate solution omits two critical ingredients. First, it invokes a direct Fourier inversion E[h((σ−nγ)/√n) ϕ(S_nx)] = (2π)^{-1}∫ ĥ(t) e^{-itγ√n} (P_{it/√n}^n ϕ)(x) dt for h=ψ_J, but ψ_J need not be in L^1, so ĥ generally has a 1/|t| singularity at 0. The paper’s main technical point is precisely to avoid this singularity using a Cauchy principal value/smoothing device (cf. discussion following Theorems 1.1–1.2) , whereas the model’s proof never installs the needed principal value or an equivalent regularization. Second, for matrix coefficients, the model replaces the hyperplane-singularity analysis by asserting a uniform-in-(n,x,y) integrability bound sup E|ι(S_nx,y)|<∞ for ι(x,y)=log(|⟨v,f⟩|/(||v|| ||f||)) and then bounding via |e^{itΔ}−1|≤|t||Δ|. This is not justified from the spectral gap alone, since |ι| is unbounded and not Hölder; the paper instead builds a partition of unity χ_k localized on distance level-sets from H_y and exploits ν(B(H_y,r))≲r^η plus finite-time large deviations to control contributions uniformly (see (4.3), Lemma 4.1, and Lemmas 4.7–4.10) . In short, while parts of the model’s outline echo the spectral method, it leaves the ξ≈0 singularity and the hyperplane-singularity uniformity unaddressed, both of which the paper resolves with new, precise devices.