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Rate of convergence of the conditioned random walk towards the Brownian bridge

Laurent Decreusefond, Antonin Jacquet

TL;DR

The paper derives quantitative rates for the convergence of two discrete processes to the Brownian bridge: a random walk conditioned to be zero at the endpoint and the empirical process from the Glivenko–Cantelli theorem. By combining a functional Stein method with a Radon–Nikodym representation of the Brownian bridge, the authors bound the Fortet–Mourier distance between these processes and the bridge, achieving a rate of order $n^{-1/18} \log n$ in the Rademacher and Poisson-minus-one cases and extending to general lattice increments via a max of local-limit and Fourier errors. The core technique discretizes the processes up to a late time $t_n$, uses RN-densities to link conditioned and unconditioned paths on $[0,t]$, and controls the remaining tail by continuity on $[t,1]$, with a set of auxiliary lemmas ensuring density convergence and Lipschitz properties. Collectively, the results provide a rigorous, general route to process-level convergence rates for conditioned and empirical path sequences toward the Brownian bridge, with explicit dependence on the increment law via $\rho(\mathcal{L},n)$ and $\tau(\mathcal{L},n,\delta)$.

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.

Rate of convergence of the conditioned random walk towards the Brownian bridge

TL;DR

The paper derives quantitative rates for the convergence of two discrete processes to the Brownian bridge: a random walk conditioned to be zero at the endpoint and the empirical process from the Glivenko–Cantelli theorem. By combining a functional Stein method with a Radon–Nikodym representation of the Brownian bridge, the authors bound the Fortet–Mourier distance between these processes and the bridge, achieving a rate of order in the Rademacher and Poisson-minus-one cases and extending to general lattice increments via a max of local-limit and Fourier errors. The core technique discretizes the processes up to a late time , uses RN-densities to link conditioned and unconditioned paths on , and controls the remaining tail by continuity on , with a set of auxiliary lemmas ensuring density convergence and Lipschitz properties. Collectively, the results provide a rigorous, general route to process-level convergence rates for conditioned and empirical path sequences toward the Brownian bridge, with explicit dependence on the increment law via and .

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.
Paper Structure (11 sections, 8 theorems, 134 equations)

This paper contains 11 sections, 8 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. General strategy
  4. Definitions and notations
  5. General notations
  6. Definitions and notations related to the distribution $\mathcal{L}$.
  7. General result
  8. Proofs
  9. Proof of Theorem \ref{['thm: theorem principal general.']}
  10. Proof of Theorem \ref{['thm: theoreme principal rademacher.']}
  11. Proof of Theorem \ref{['thm: theoreme principal poisson moins 1.']}

Key Result

Theorem 2.1

Assume that $\mathcal{L}$ is the Rademacher distribution, i.e. Then, there exists a constant $C>0$ such that for all $n \ge 1$, where $\mu_n(\mathcal{L})$ is the distribution of $S_{2n}$ conditioned to be $0$ at time $1$, $S_{2n}$ is the process defined at eq: definition de Sn. and $B^{br}$ is the Brownian bridge.

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 8 more