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Asymptotics for additive functionals of particle systems via Stein's method

Arturo Jaramillo, Antonio Murillo-Salas

TL;DR

The paper develops a Stein–Mecke–Mecke-based third-moment framework for Gaussian fluctuations of additive functionals in Poisson-driven, measure-valued particle systems. It provides explicit Wasserstein-distance bounds and extends central limit-type results to a wide class of dynamics, including α-stable motions, diffusions, and fractional Brownian motions, with concrete rates. The Dyson Brownian motion case is treated as a key spectral-example, illustrating how Poisson input and non-Markovian dynamics can still yield quantitative normal approximations. Collectively, the results offer a unified method to obtain rates of convergence for a broad family of evolving measure-valued systems, bridging occupation-time fluctuations and spectral empirical measures. The approach yields new quantitative CLTs in settings where classical Markovian techniques are not applicable and broadens the scope of Gaussian fluctuation analysis for interacting particle systems.

Abstract

We consider additive functionals of systems of random measures whose initial configuration is given by a Poisson point process, and whose individual components evolve according to arbitrary Markovian or non-Markovian measure valued dynamics, with no structural assumptions beyond basic moment bounds. In this setting and under adequate conditions, we establish a general third moment theorem for the normalized functionals. Building on this result, we obtain the first quantitative bounds in the Wasserstein distance for a variety of moving-measure models initialized by Poisson-driven clouds of points, turning qualitative central limit theorems into explicit rates of convergence. The scope of the approach is then demonstrated through several examples, including systems driven by fractional Brownian motion, $α$-stable processes, uniformly elliptic diffusions, and spectral empirical measures arising from Dyson Brownian motion, all under broad assumptions on the control measure of the initial Poisson configuration. The analysis relies on a combination of Stein's method with Mecke's formula, in the spirit of the Poisson Malliavin-Stein methodology.

Asymptotics for additive functionals of particle systems via Stein's method

TL;DR

The paper develops a Stein–Mecke–Mecke-based third-moment framework for Gaussian fluctuations of additive functionals in Poisson-driven, measure-valued particle systems. It provides explicit Wasserstein-distance bounds and extends central limit-type results to a wide class of dynamics, including α-stable motions, diffusions, and fractional Brownian motions, with concrete rates. The Dyson Brownian motion case is treated as a key spectral-example, illustrating how Poisson input and non-Markovian dynamics can still yield quantitative normal approximations. Collectively, the results offer a unified method to obtain rates of convergence for a broad family of evolving measure-valued systems, bridging occupation-time fluctuations and spectral empirical measures. The approach yields new quantitative CLTs in settings where classical Markovian techniques are not applicable and broadens the scope of Gaussian fluctuation analysis for interacting particle systems.

Abstract

We consider additive functionals of systems of random measures whose initial configuration is given by a Poisson point process, and whose individual components evolve according to arbitrary Markovian or non-Markovian measure valued dynamics, with no structural assumptions beyond basic moment bounds. In this setting and under adequate conditions, we establish a general third moment theorem for the normalized functionals. Building on this result, we obtain the first quantitative bounds in the Wasserstein distance for a variety of moving-measure models initialized by Poisson-driven clouds of points, turning qualitative central limit theorems into explicit rates of convergence. The scope of the approach is then demonstrated through several examples, including systems driven by fractional Brownian motion, -stable processes, uniformly elliptic diffusions, and spectral empirical measures arising from Dyson Brownian motion, all under broad assumptions on the control measure of the initial Poisson configuration. The analysis relies on a combination of Stein's method with Mecke's formula, in the spirit of the Poisson Malliavin-Stein methodology.

Paper Structure

This paper contains 20 sections, 9 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Model description
  3. Poisson random parameter space
  4. Measure valued processes
  5. Additive functionals
  6. Fundamental examples
  7. Preliminaries
  8. Stein's method
  9. Mecke's formula
  10. Abstract Moment Theorems
  11. Application to Particle Systems
  12. Particles performing $\alpha$-stable motions
  13. Particles moving as a diffusion on $\mathbb{R}^d$
  14. Particles moving as a fractional Brownian motion on $\mathbb{R}^d$
  15. Application to the Dyson process
  16. ...and 5 more sections

Key Result

Theorem 1

We have that Moreover, under the condition $\psi\geq 0$, the following third moment theorem holds

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1
  • Remark 2
  • proof : Proof of Theorem \ref{['thm:mainabstractone']}
  • Theorem 2
  • Proposition 5.1
  • Proposition 5.2
  • Remark 3
  • Proposition 5.3
  • Proposition 6.1
  • ...and 6 more