Edgeworth expansion on Wiener chaos

TL;DR

The paper delivers a full Edgeworth-type expansion for functionals $F$ in a fixed Wiener chaos $\\mathcal{W}_p$, quantifying the distance to a Gaussian law corrected by Hermite polynomials up to order $4m-1$. The main finding provides a total variation bound $d_{TV}(\\mathbb{P}_F, \\boldsymbol{\\gamma}_{F,m}) \le C_{p,m} (\\operatorname{Var}(\\Gamma(F,F)))^{(m+1)/2}$, with the density correction given by $x \mapsto \\frac{e^{-x^2/2}}{\\sqrt{2\\pi}} \\left(1+ \\sum_{k=3}^{4m-1} \\frac{\\mathbb{E}[H_k(F)]}{k!} H_k(x) \\right)$. The approach is Malliavin–Stein-based, harnessing the carré-du-champ $\\Gamma$ and the Ornstein–Uhlenbeck structure to perform higher-order inversions and integration-by-parts, yielding explicit remainder bounds that improve with the expansion order. This framework not only recovers the optimal fourth-moment bound as a byproduct but also extends the methodology to a general Markov-diffusive setting, enabling Edgeworth-type refinements in broader stochastic chaos contexts and under matching-moment conditions. The results have potential implications for refined Gaussian approximations in stochastic analysis, random matrix theory, and diffusion-based models where chaos decompositions govern fluctuations.

Abstract

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 $\boldsymbolγ_{F,m}$ be the Radon measure with density $x \mapsto \frac{e^{-x^2/2}}{\sqrt{2π}} \left(1 + \sum_{k=3}^{4m-1} \frac{\E[H_k(F)]}{k!}\, H_k(x)\right)$, 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 $\boldsymbolγ_{F,m}$ is of order $\Var(Γ(F,F))^{({m+1})/{2}}$, where $Γ(F,F)$ denotes the carré-du-champ operator of $F$. The variance of $Γ(F,F)$ is known to govern Gaussian fluctuations and can be bounded from above by $κ_4(F)$, 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 $\Var(Γ(F,F))$ 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.

Figures (4)

• Figure 1: Simulated densities (first column) and their first and second derivatives (second and third columns) for the Gaussian distribution (red), the KDE estimate (blue), and the Edgeworth expansion from Theorem \ref{['maintheorem']} (green), in the case where $F$ is a fractional Brownian motion as described in Section \ref{['sec:hermvar']}.
• Figure 2: Simulated densities (first column) and their first and second derivatives (second and third columns) for the Gaussian distribution (red), the KDE estimate (blue), and the Edgeworth expansion from Theorem \ref{['maintheorem']} (green), with $F$ as specified in Section \ref{['sec:goe']}.
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Theorems & Definitions (145)

• Definition 1.1: Edgeworth expansion, general form
• Theorem 1.2
• Proposition 1.3
• Lemma 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5