[Paper Review] Stein's method for dependent random variables occurring in Statistical Mechanics

Peter Eichelsbacher, Matthias Löwe|ArXiv.org|Aug 13, 2009

Random Matrices and Applications11 references10 citations

TL;DR

This paper develops Stein's method for exchangeable pairs to establish optimal Berry-Esseen bounds for partial sums of dependent random variables in Curie-Weiss models of statistical mechanics. It proves convergence rates to non-Gaussian limit distributions with densities proportional to $\exp(-\mu|x|^{2k}/(2k)!)$, including the optimal $O(n^{-1/2})$ rate in the central limit theorem at critical temperature $\beta_c = 1$, even when the CLT fails.

ABSTRACT

We obtain rates of convergence in limit theorems of partial sums $S_n$ for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the Curie-Weiss models. Under appropriate assumptions there exists a real number $α$, a positive real number $μ$, and a positive integer $k$ such that $(S_n- n α)/n^{1 - 1/2k}$ converges weakly to a random variable with density proportional to $\exp(-μ|x|^{2k} /(2k)!)$. We develop Stein's method for exchangeable pairs for a rich class of distributional approximations including the Gaussian distributions as well as the non-Gaussian limit distributions with density proportional to $\exp(-μ|x|^{2k} /(2k)!)$. Our results include the optimal Berry-Esseen rate in the Central Limit Theorem for the total magnetization in the classical Curie-Weiss model, for high temperatures as well as at the critical temperature $β_c=1$, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature $1/ β_n$ converges to one and obtain a threshold for the speed of this convergence. Single spin distributions satisfying the Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or continuous Curie-Weiss models are considered.

Motivation & Objective

• To extend Stein’s method to exchangeable pairs for non-Gaussian limit distributions arising in statistical mechanics.
• To establish sharp rates of convergence for partial sums of dependent, identically distributed random variables in Curie-Weiss models.
• To analyze the behavior of convergence rates as the inverse temperature $\beta_n \to 1$, the critical temperature.
• To provide a rigorous framework for Berry-Esseen bounds in models satisfying the Griffiths-Hurst-Sherman (GHS) inequality.
• To derive optimal convergence rates in the central limit theorem for the total magnetization, even at criticality where standard CLT fails.

Proposed method

• Adapts Stein’s method for exchangeable pairs to a class of non-Gaussian target distributions with densities proportional to $\exp(-\mu|x|^{2k}/(2k)!)$.
• Uses Stein’s identity and solution of the Stein equation for distributions with densities $p(x) \propto \exp(-a_k V(x))$, where $V(x) = |x|^{2k}$.
• Establishes bounds on the solution $f_h$ of the Stein equation via integration by parts and moment estimates, ensuring regularity conditions for the method.
• Applies the method to Curie-Weiss models with spins distributed under $P_{n,\beta,h}$, where the total magnetization $S_n$ is the key sum.
• Derives bounds on $\mathbb{E}|f_h(X) - f_h(Y)|$ for exchangeable pairs $(X,Y)$ constructed via Gibbs sampling or local move dynamics.
• Uses the GHS inequality to ensure the required convexity and symmetry properties of the potential function $V(x)$, enabling moment and tail estimates.

Experimental results

Research questions

• RQ1 What is the optimal rate of convergence in the central limit theorem for the total magnetization in the Curie-Weiss model at the critical temperature $\beta_c = 1$?
• RQ2 How does the convergence rate behave as the inverse temperature $\beta_n \to 1$ from above or below?
• RQ3 Can Stein’s method be extended to non-Gaussian limit distributions with heavy-tailed or higher-order polynomial tails, such as $\exp(-\mu|x|^{2k}/(2k)!)$?
• RQ4 What are the conditions under which the classical central limit theorem fails for dependent random variables in mean-field models?
• RQ5 How can exchangeable pairs be constructed in Curie-Weiss models to enable Stein’s method for dependent sequences?

Key findings

• The paper establishes the optimal Berry-Esseen rate of $O(n^{-1/2})$ for the total magnetization in the Curie-Weiss model at the critical temperature $\beta_c = 1$, where the classical CLT fails.
• For $k \geq 2$, the normalized sum $(S_n - n\alpha)/n^{1-1/2k}$ converges weakly to a non-Gaussian limit with density proportional to $\exp(-\mu|x|^{2k}/(2k)!)$, and the convergence rate is $O(n^{-1/2k})$.
• The convergence rate is shown to be optimal in the sense that no faster rate is possible under the given dependence structure.
• As $\beta_n \to 1$, the rate of convergence to the non-Gaussian limit is controlled by a threshold depending on the speed of $\beta_n$ approaching 1, with the rate deteriorating as $\beta_n$ approaches 1 from above.
• The method applies to models satisfying the GHS inequality, such as continuous Curie-Weiss models and liquid helium models, ensuring the required convexity and symmetry for the potential function.
• The solution $f_h$ to the Stein equation for the non-Gaussian target distribution satisfies uniform bounds on $f_h$, $f_h'$, and $f_h''$, which are essential for bounding the error in the Stein method.