[Paper Review] Stein's method for Brownian approximations

Laure Coutin, Laurent Decreusefond|arXiv (Cornell University)|Jul 15, 2012

Random Matrices and Applications7 references3 citations

TL;DR

This paper develops a Stein method framework for Wasserstein distance approximation in Hilbert spaces to analyze the convergence rate of Brownian motion approximations via Poisson processes and Donsker's theorem. By combining Malliavin calculus with Stein's method on spaces of Hölder-continuous functions, it establishes that the convergence rate for Poisson approximation of Brownian motion is $\lambda^{-1/2}$, resolving a gap caused by path discontinuities in prior approaches.

ABSTRACT

Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for the Poisson approximation of the Brownian motion is as expected proportional to $\\lambda^{-1/2}$ where $\\lambda$ is the intensity of the Poisson process. We also exhibit the speed of convergence for the Donsker Theorem and for the linear interpolation of the Brownian motion. By iterating the procedure, we give Edgeworth expansions with precise error bounds.

Motivation & Objective

• To develop a Stein method framework for bounding Wasserstein distances between probability measures on infinite-dimensional Hilbert spaces.
• To resolve the discrepancy in convergence rates caused by path discontinuities in Poisson processes when approximating Brownian motion.
• To establish the speed of convergence for Donsker's theorem and linear interpolation of Brownian motion using a refined approximation framework.
• To extend Stein's method beyond finite-dimensional Gaussian approximations by incorporating Malliavin calculus for Hilbert-space-valued processes.

Proposed method

• Formulates a Stein method on Hilbert spaces using two types of Malliavin gradients: one for characterizing the target Gaussian measure and one for the approximating process.
• Applies integration by parts formulas from Malliavin calculus to bypass the need for explicit couplings in Stein's method.
• Uses the space $\mathcal{I}_{\beta,2}$ for $\beta < 1/2$ to embed both piecewise differentiable Poisson paths and Hölder-continuous Brownian paths.
• Derives bounds on the Wasserstein distance via the solution of the Stein equation involving $1$-Lipschitz functions and their second-order derivatives.
• Employs the transfer principle to extend results from standard Brownian motion to fractional Brownian motion via continuous linear operators.
• Controls error terms via operator norms and fractional integral operators, particularly $I^{1-\beta}_{0^+}$, to derive convergence rates.

Experimental results

Research questions

• RQ1 What is the convergence rate of the normalized compensated Poisson process to Brownian motion in the Wasserstein distance?
• RQ2 How can the Stein method be adapted to infinite-dimensional Hilbert spaces to analyze distributional convergence?
• RQ3 Can the path discontinuity issue in Poisson-Brownian approximation be resolved through a common function space framework?
• RQ4 What is the speed of convergence for the Donsker theorem when using linear interpolation of random walks?
• RQ5 How does the convergence rate depend on the regularity of the limiting process, such as fractional Brownian motion?

Key findings

• The convergence rate for the Poisson approximation of Brownian motion is $\lambda^{-1/2}$, confirming the expected rate despite prior suggestions of a corrective term.
• The framework resolves the path discontinuity gap by embedding both processes in the space $\mathcal{I}_{\beta,2}$ for $\beta < 1/2$, ensuring common regularity.
• For Donsker's theorem, the convergence rate is shown to be $\lambda^{-1/2}$ when using linear interpolation of symmetric random walks.
• The method achieves a convergence rate of $\frac{a}{3\sqrt{\lambda}}$ for the approximation of fractional Brownian motion of Hurst index $H$ by Poisson chaos.
• The error bound is controlled via the norm $\sum_{j=1}^{m}\|H_{m}^{\sharp}(j)\|_{l^2(\mathbb{N})}^3 \leq \frac{m^{3\beta - 7/2}}{(1-\beta)\Gamma(1-\beta)^2}$, which dominates the convergence rate.
• The transfer principle allows extension of results to fractional Brownian motion via the operator $\Theta_H = K_H \circ I^{-1}_{0^+}$, preserving convergence rates.