Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extension of a theorem of Wschebor to free and matrix Brownian motions

Catherine Donati-Martin, Alain Rouault

TL;DR

This work extends Wschebor’s law of large numbers and the associated fluctuation theory from scalar Brownian motion to two non-scalar frontiers: free Brownian motion and matrix-valued Brownian motion. It develops LLNs in the free setting toward semicircular distributions and in the matrix setting toward GUE moments, while establishing fluctuation theorems via Chebyshev and Hermite trace polynomials, and matrix-variate polynomial bases. The paper introduces a matrix-probability framework including Hermite trace polynomials and matrix-variate Hermite polynomials, and proves a Wiener-Wigner transfer principle to connect scalar and free fluctuations. In the large-N limit, the results recover free-probability-like limits with Catalan-number combinatorics, and a detailed asymptotic analysis clarifies how matrix fluctuations converge to isotropic Gaussian matrices with spectral distributions governed by semicircular laws. Overall, the work bridges classical LLN/CLT phenomena for smoothing of Brownian paths with noncommutative and random-matrix models, offering explicit fluctuation descriptions and combinatorial structures that unify scalar, free, and matrix settings.

Abstract

In 1992, M. Wschebor proved a theorem on the convergence of small increments of the Brownian motion. Since then, it has been extended to various processes. We prove a version of this theorem for the Hermitian Brownian motion and the free Brownian motion. Since these theorems deal with a convergence to a deterministic limit, we prove also the convergence in distribution of the corresponding fluctuations.

Extension of a theorem of Wschebor to free and matrix Brownian motions

TL;DR

This work extends Wschebor’s law of large numbers and the associated fluctuation theory from scalar Brownian motion to two non-scalar frontiers: free Brownian motion and matrix-valued Brownian motion. It develops LLNs in the free setting toward semicircular distributions and in the matrix setting toward GUE moments, while establishing fluctuation theorems via Chebyshev and Hermite trace polynomials, and matrix-variate polynomial bases. The paper introduces a matrix-probability framework including Hermite trace polynomials and matrix-variate Hermite polynomials, and proves a Wiener-Wigner transfer principle to connect scalar and free fluctuations. In the large-N limit, the results recover free-probability-like limits with Catalan-number combinatorics, and a detailed asymptotic analysis clarifies how matrix fluctuations converge to isotropic Gaussian matrices with spectral distributions governed by semicircular laws. Overall, the work bridges classical LLN/CLT phenomena for smoothing of Brownian paths with noncommutative and random-matrix models, offering explicit fluctuation descriptions and combinatorial structures that unify scalar, free, and matrix settings.

Abstract

In 1992, M. Wschebor proved a theorem on the convergence of small increments of the Brownian motion. Since then, it has been extended to various processes. We prove a version of this theorem for the Hermitian Brownian motion and the free Brownian motion. Since these theorems deal with a convergence to a deterministic limit, we prove also the convergence in distribution of the corresponding fluctuations.

Paper Structure

This paper contains 23 sections, 24 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. The scalar case
  3. LLN . Proof of Theorem \ref{['W0']}.
  4. Fluctuations
  5. The free case
  6. LLN
  7. Fluctuations
  8. Matricial case : LLN
  9. Matricial case : fluctuations
  10. Hermite polynomials for Hermitian Brownian motion
  11. Fluctuations
  12. Asymptotics $N \rightarrow \infty$
  13. The coefficients $K_{k,n,N}$
  14. The coefficents $a_{n,N}$ and $b_{n,N}$
  15. Matrix-variate Hermite polynomials
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\varphi : \mathbb R \to \mathbb R$ with compact support and bounded variation satisfying $\Vert\varphi\Vert_2 = 1$ and and let be the corresponding occupation measure. Then almost surely as $\varepsilon \to 0$, $\mu_\varepsilon$ converges to the standard normal distribution $\mathcal{N}(0,1)$ for the convergence of moments, i.e. that for every $k \geq 0$ where $\mathcal{N} \mathrel{\mathop{

Theorems & Definitions (32)

  • Theorem 1.1: wschebor1992accroissements, azais1996almost, wschebor2006smoothing
  • Theorem 1.2
  • Corollary 2.1
  • Remark 1
  • Remark 2
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Proposition 4.1
  • ...and 22 more