Edgeworth expansion on Wiener chaos
Consider $F$ an element of the $p$-th Wiener chaos $\WW_p$, and denote by $\prob_F$ its law. For a positive integer $m$, let $\boldsymbol{\gamma}_{F,m}$ be the Radon measure with density $x \mapsto \frac{e^{-x^2/2}}{\sqrt{2\pi}} \left(1 + \sum_{k=3}^{4m-1} \frac{\E[H_k(F)]}{k!}\, H_k(x)\right)$, where $H_k$ is the $k$-th Hermite polynomial. The main goal of this article is to prove that the total variation distance between $\prob_F$ and $\boldsymbol{\gamma}_{F,m}$ is of order $\Var(\Gamma(F,F))^{({m+1})/{2}}$, where $\Gamma(F,F)$ denotes the carr\'e-du-champ operator of $F$. The variance of $\Gamma(F,F)$ is known to govern Gaussian fluctuations and can be bounded from above by $\kappa_4(F)$, the fourth cumulant of $F$, as established in the seminal work \cite{NP2009a}. Our result thus provides a genuine Edgeworth expansion in the setting of central convergence on Wiener chaoses. In this context, the quantity $\Var(\Gamma(F,F))$ plays the role of the small parameter that governs the accuracy of the approximation, in the same way that $1/\sqrt{n}$ does in the classical central limit theorem. To the best of our knowledge, our work is the first to establish Edgeworth expansions for Wiener chaoses in full generality and at arbitrary order, together with explicit remainder bounds that systematically improve with the order of the expansion--exactly as one would expect from an Edgeworth approximation. Our results apply verbatim to every situation where a central limit theorem is available for chaos elements, since no structural assumption is required beyond belonging to a fixed Wiener chaos. As a byproduct, we recover the celebrated optimal fourth moment theorem from \cite{NP2015} by combining the expansions at the first and second orders, with sharper quantitative bounds. Previous works on Edgeworth expansions for Wiener chaoses were essentially restricted to the first order.