Sharp Fourth-Moment Wasserstein CLT Bounds
Abstract
We prove that for the normalized sum W=∑i=1n Xi/√n of independent, mean-zero, d-dimensional random vectors with Cov(W) =Σ=diag(σ12,…, σd2) (the diagonal form of the covariance matrix is assumed without loss of generality), the 2-Wasserstein distance from W to N(0,Σ) is bounded by √(2∑j=1d min {σj2, (∑i=1n 𝔼 [|Xi'−Xi|2(X'ij−Xij)2 ] )/(n2 σj2) }), where |·| denotes the Euclidean norm, Xi' is an independent copy of Xi, and Xij (Xij' resp.) denotes the jth coordinate of the vector Xi (Xi' resp.). On the one hand, it gives the optimal rate O(β√(d/n)) if |Xi|≤ β for all i. On the other hand, in the isotropic case where Σ=Id, it gives the optimal fourth-moment bound O(√(∑i=1n 𝔼|Xi|4/n2)). Moreover, the coordinatewise minima suggest a tradeoff between the naïve coupling and the normal coupling for each component. To prove the main result, we develop a new transportation inequality using a coupling via the Föllmer process with a general covariance matrix and the martingale property of its drift. The resulting bound is then controlled using a Bonis-type expansion together with Bessel's inequality. The method and the result extend to p-Wasserstein bounds. As a consequence, we obtain the Hilbert space 4-Wasserstein CLT bound O(n−1/4[tr(Q1/2)]1/2), where, in the i.i.d. case, Q=𝔼[|X1'−X1|2(X1'−X1)⊗2] and the O(n−1/4) rate is optimal given the dimension-free form of Q. We further strengthen it to a dimension-free strong approximation result with a near-optimal error rate through the KMT embedding construction.
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BibTeX
@misc{fang2026sharp,
title = {{Sharp Fourth-Moment Wasserstein CLT Bounds}},
author = {Xiao Fang and Zi-Yao Su and Jiaxue Wang and Xiaolin Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35354},
eprint = {2610.35354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35354},
}
RIS
TY - UNPB
TI - Sharp Fourth-Moment Wasserstein CLT Bounds
AU - Xiao Fang
AU - Zi-Yao Su
AU - Jiaxue Wang
AU - Xiaolin Wang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35354
UR - https://arxiv.org/abs/2610.35354
AB - We prove that for the normalized sum W=∑_i=1^n X_i/√n of independent, mean-zero, d-dimensional random vectors with Cov(W) =Σ=diag(σ_1^2,…, σ_d^2) (the diagonal form of the covariance matrix is assumed without loss of generality), the 2-Wasserstein distance from W to N(0,Σ) is bounded by √(2∑_j=1^d min {σ_j^2, (∑_i=1^n 𝔼 [|X_i'−X_i|^2(X'_ij−X_ij)^2 ] )/(n^2 σ_j^2) }), where |·| denotes the Euclidean norm, X_i' is an independent copy of X_i, and X_ij (X_ij' resp.) denotes the jth coordinate of the vector X_i (X_i' resp.). On the one hand, it gives the optimal rate O(β√(d/n)) if |X_i|≤ β for all i. On the other hand, in the isotropic case where Σ=I_d, it gives the optimal fourth-moment bound O(√(∑_i=1^n 𝔼|X_i|^4/n^2)). Moreover, the coordinatewise minima suggest a tradeoff between the naïve coupling and the normal coupling for each component. To prove the main result, we develop a new transportation inequality using a coupling via the Föllmer process with a general covariance matrix and the martingale property of its drift. The resulting bound is then controlled using a Bonis-type expansion together with Bessel's inequality. The method and the result extend to p-Wasserstein bounds. As a consequence, we obtain the Hilbert space 4-Wasserstein CLT bound O(n^−1/4[tr(Q^1/2)]^1/2), where, in the i.i.d. case, Q=𝔼[|X_1'−X_1|^2(X_1'−X_1)^⊗2] and the O(n^−1/4) rate is optimal given the dimension-free form of Q. We further strengthen it to a dimension-free strong approximation result with a near-optimal error rate through the KMT embedding construction.
ER -