Optimal Sobolev Rate for Gaussian Density Approximation of Wiener Chaos Vectors
Abstract
Let (Fn) be a sequence of random vectors with identity covariance matrix whose components belong to the same fixed Wiener chaos, and assume that Fn converges in law to a standard Gaussian vector. We prove that, for every integer m≥0 and every p∈[1,∞], the optimal rate of convergence of the densities in the Sobolev space Wm,p is given by the maximum of the absolute third-order cumulants and the diagonal fourth-order cumulants. The same quantity also gives the optimal rates in total variation, Kolmogorov and $1$-Wasserstein distances. Our proof first derives, by Gaussian interpolation and Gaussian convolution, a cumulant expansion in the space of tempered distributions without imposing Malliavin nondegeneracy at the endpoint. Finite-order Malliavin density estimates then upgrade this identity to Sobolev spaces. Matching lower bounds follow from a finite-dimensional argument in which parity separates the third-order and fourth-order Gaussian corrections, while norm equivalence rules out cancellations among mixed cumulants. A superconvergence theorem supplies the required finite negative moments of the Malliavin determinant along a sufficiently far tail of the approximating sequence, so that no Malliavin nondegeneracy assumption is required.
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BibTeX
@misc{chen2026optimal,
title = {{Optimal Sobolev Rate for Gaussian Density Approximation of Wiener Chaos Vectors}},
author = {Huiping Chen and Yong Chen and Yong Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07066},
eprint = {2609.07066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07066},
}
RIS
TY - UNPB
TI - Optimal Sobolev Rate for Gaussian Density Approximation of Wiener Chaos Vectors
AU - Huiping Chen
AU - Yong Chen
AU - Yong Liu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07066
UR - https://arxiv.org/abs/2609.07066
AB - Let (F_n) be a sequence of random vectors with identity covariance matrix whose components belong to the same fixed Wiener chaos, and assume that F_n converges in law to a standard Gaussian vector. We prove that, for every integer m≥0 and every p∈[1,∞], the optimal rate of convergence of the densities in the Sobolev space W^m,p is given by the maximum of the absolute third-order cumulants and the diagonal fourth-order cumulants. The same quantity also gives the optimal rates in total variation, Kolmogorov and $1$-Wasserstein distances. Our proof first derives, by Gaussian interpolation and Gaussian convolution, a cumulant expansion in the space of tempered distributions without imposing Malliavin nondegeneracy at the endpoint. Finite-order Malliavin density estimates then upgrade this identity to Sobolev spaces. Matching lower bounds follow from a finite-dimensional argument in which parity separates the third-order and fourth-order Gaussian corrections, while norm equivalence rules out cancellations among mixed cumulants. A superconvergence theorem supplies the required finite negative moments of the Malliavin determinant along a sufficiently far tail of the approximating sequence, so that no Malliavin nondegeneracy assumption is required.
ER -