Gaussian approximation on the Skorokhod space via Malliavin calculus and regularization
Solesne Bourguin, Simon Campese
TL;DR
This work develops a Banach-space-valued Malliavin calculus framework to quantify Gaussian approximation for Banach-valued random elements in the Skorokhod space. It introduces a Banach-valued carré du champ Γ, valued in the projective tensor product, and proves integration-by-parts formulas that bound |E f(X) - E f(Z)| in terms of Γ and the Gaussian covariance, supplemented by a regularization procedure to bound the bounded Lipschitz distance. The paper then provides contraction bounds for Wiener chaos, enabling explicit control of the Γ-term via kernel contractions, and applies these results to a Gaussian-subordinated Hermite model to obtain concrete rates. A Hilbert-space specialization recovers and extends known results, illustrating the broad applicability of the approach to functional Gaussian approximation in infinite dimensions. Overall, the work offers a cohesive, dimension-free toolkit for quantitative Gaussian approximation in Banach spaces with explicit contraction-based controls.
Abstract
We introduce a carré du champ operator for Banach-valued random elements, taking values in the projective tensor product, and use it to control the bounded Lipschitz distance between a Malliavin-smooth random element satisfying mild regularity assumptions and a Radon Gaussian taking values in the Skorokhod space equipped with the uniform topology. In the case where the random element is a Banach-valued multiple integral, the carré du champ expression is further bounded by norms of the contracted integral kernel. The main technical tool is an integration by parts formula, which might be of independent interest. As a by-product, we recover a bound obtained recently by Düker and Zoubouloglou in the Hilbert space setting and complement it by providing contraction bounds.