[Paper Review] A new quantitative central limit theorem on the Wiener space with applications to Gaussian processes

Tobias Fissler, Christoph Thaele|arXiv (Cornell University)|Oct 5, 2016

Stochastic processes and financial applications3 citations

TL;DR

This paper establishes a new quantitative central limit theorem on the Wiener space for non-linear functionals of Gaussian processes, using contraction norms of chaos kernels to bound distances to normality. It reveals a novel link between the Hermite rank of a function and the 'chaotic gap,' showing universal convergence rates for power variations of fractional Brownian motion regardless of the moment order p.

ABSTRACT

Quantitative limit theorems for non-linear functionals on the Wiener space are considered. Given the possibly infinite sequence of kernels of the chaos decomposition of such a functional, an estimate for different probability distances between the functional and a Gaussian random variable in terms of contraction norms of these kernels is derived. The applicability of this result is demonstrated by means of the Breuer-Major theorem, unfolding thereby a new connection between the Hermite rank of the considered function and a chaotic gap. Especially, power variations of the fractional Brownian motion and processes belonging to the Cauchy class are studied.

Motivation & Objective

• To develop a quantitative central limit theorem for non-linear functionals on the Wiener space that applies to infinite chaos decompositions.
• To address the limitation of existing methods—like the second-order Poincaré inequality—that yield suboptimal convergence rates in infinite chaos settings.
• To provide an analytical bound on probability distances (e.g., Wasserstein) between a functional and a Gaussian variable using only the kernels of its Wiener chaos decomposition.
• To demonstrate the applicability of the method through the Breuer-Major theorem and power variation of fractional Brownian motion.
• To uncover a new structural relationship between the Hermite rank of a function and the chaotic gap in the convergence rate.

Proposed method

• The method uses Malliavin-Stein techniques combined with Wiener chaos decomposition to express a functional $ F = igoplus_{q=0}^ au I_q(f_q) $ as an infinite sum of multiple stochastic integrals.
• It introduces contraction norms of the kernels $ f_q $, specifically $ ig floor f_p ilde{igotimes}_r f_q ig floor_{p+q-2r} $, to bound the Wasserstein distance between $ F $ and a standard Gaussian variable.
• The key inequality bounds the Wasserstein distance $ d_{mW}( extbf{F}, extbf{Z}) $ via a sum over all pairs of chaos orders $ p, q $, involving factorials, binomial coefficients, and contraction norms.
• The method distinguishes between diagonal terms ($ p = q $) and off-diagonal terms ($ p eq q $), with separate expressions for the contraction contributions.
• It applies the theory to the Breuer-Major setting, where $ F_n = rac{1}{ ilde{n}} igsum_{k=1}^n ig( g(X_k) - ext{E}[g(X_k)] ig) $, with $ X $ a stationary Gaussian process.
• The approach reveals that the convergence rate depends on the interplay between the Hermite rank of $ g $ and the chaotic gap, defined as the minimal $ r $ such that $ f_r eq 0 $.

Experimental results

Research questions

• RQ1 Can a quantitative central limit theorem be established for functionals in the infinite Wiener chaos decomposition using only the kernel norms?
• RQ2 How does the Hermite rank of a function $ g $ influence the convergence rate in the central limit theorem for power variations of Gaussian processes?
• RQ3 What is the role of the 'chaotic gap'—the first non-zero chaos order—in determining the rate of convergence?
• RQ4 Does the convergence rate for power variations of fractional Brownian motion depend on the moment order $ p $, or is it universal?
• RQ5 Can the Malliavin-Stein method be extended to yield optimal, analytical bounds in infinite chaos settings without relying on Poincaré-type inequalities?

Key findings

• The paper derives a new quantitative bound on the Wasserstein distance between a functional $ F $ on the Wiener space and a standard Gaussian variable $ Z $, expressed in terms of contraction norms of its Wiener chaos kernels.
• The bound is given by $ d_{mW}( extbf{F}, extbf{Z}) riangleq ext{sup}_{h ext{ Lipschitz}} | ext{E}[h(F)] - ext{E}[h(Z)]| riangleq c ig( ext{sum over } p,q,r ext{ of terms involving } (r-1)! {p-1 race r-1}^2 ext{ and } ig floor f_p ilde{igotimes}_r f_q ig floor_{p+q-2r} ig) $, with explicit coefficients.
• For power variations of fractional Brownian motion with $ g(x) = |x|^p - ext{E}[|X_1|^p] $, the convergence rate is universal and independent of $ p $, matching the known rate for quadratic variation ($ p=2 $).
• The convergence rate is governed by the interplay between the Hermite rank of $ g $ and the chaotic gap: a higher Hermite rank or a larger chaotic gap leads to faster convergence.
• The method achieves optimal rates in the Breuer-Major setting, avoiding the suboptimal bounds of the second-order Poincaré inequality.
• The result extends to the multivariate case, providing a Wasserstein distance bound for vector-valued functionals via a sum over all component pairs and chaos orders.