[论文解读] Rate of convergence of the conditioned random walk towards the Brownian bridge

Laurent Decreusefond, Antonin Jacquet|arXiv (Cornell University)|Jan 16, 2026

Random Matrices and Applications被引用 0

一句话总结

本文给出两种离散化过程（在时间 2n 时条件为零的随机游走与经验过程）向布朗桥收敛的速率的 Fortet–Mourier 距离，通过函数 Stein 方法与 Radon–Nikodym 表示实现。

ABSTRACT

We study the rate of convergence of two discrete processes towards the Brownian bridge: the random walk conditioned to be zero at time 2n and the empirical process which appears in the Glivencko-Cantelli theorem. Combining a functional Stein method with a Radon-Nikodym representation of the bridge, we bound the Fortet-Mourier distance between these conditioned processes and the Brownian bridge.

研究动机与目标

• Motivate the study of convergence rates of conditioned discrete processes to the Brownian bridge.
• Develop and apply a functional Stein method combined with Radon–Nikodym representations to bound distances to the bridge.
• Obtain explicit rate bounds in Fortet–Mourier distance for specific and general lattice-distributed increments.
• Link conditioned random walk and empirical process convergence to the Brownian bridge under a unified framework.

提出的方法

• Use a functional Stein approach to characterize the Brownian bridge as a target distribution.
• Represent conditional laws via Radon–Nikodym derivatives that are Lipschitz functions of the path.
• Discretize processes and relate distance bounds to Lipschitz test functionals.
• Introduce a time-splitting strategy choosing a time t depending on n to handle singularities near time 1.
• Derive rate bounds involving n^{-1/18} up to logarithmic factors in Fortet–Mourier distance.
• Extend results from Poisson-like conditioned processes to lattice-distributed increments with a κ(L) factor.

实验结果

研究问题

• RQ1 What is the rate of Fortet–Mourier convergence of the random walk conditioned to be zero at time 1 to the Brownian bridge?
• RQ2 What is the rate for the empirical process (Glivenko–Cantelli context) toward the Brownian bridge?
• RQ3 How do lattice-distribution increments affect the convergence rate and how can Radon–Nikodym representations be leveraged?
• RQ4 Can a unified approach cover both conditioned random walks and empirical processes to the Brownian bridge under general lattice laws?
• RQ5 What are the precise error terms and conditions (δ, ρ) that govern the rate bounds?

主要发现

• For the Rademacher case, the Fortet–Mourier distance to the Brownian bridge is bounded by C n^{-1/18} log(n).
• For the empirical process, the Fortet–Mourier distance to the Brownian bridge is bounded by C n^{-1/18} log(n).
• For the Poisson-minus-one (continuous Poisson walk) conditioned to be zero at time 1, the Fortet–Mourier distance to the Brownian bridge is bounded by C n^{-1/18} log(n).
• The general lattice-distribution result yields a bound of the form dist_F.M.(μ_n(L), B^br) ≤ C max(n^{-1/18} log n, ρ(L,n), τ(L,n,δ)).
• Convergence rates are affected by the singularity at time 1 and the absolute continuity considerations between bridge and Brownian motion.