Mean-Field Asymptotic Regime

• Mean-Field Asymptotic Regime is a framework where the collective effect of numerous interacting particles converges to a deterministic average behavior.
• It simplifies the analysis of complex systems by approximating individual interactions with a deterministic McKean–Vlasov limit.
• Key methodologies such as propagation-of-chaos and LAN analysis ensure robust statistical inference with verifiable identifiability and efficiency.

The mean-field asymptotic regime describes the limiting behavior of systems composed of a large number of interacting components, often referred to as particles, agents, or nodes, as the system size $N \to \infty$. In this regime, each component interacts with the "average" effect of the entire system rather than with individual components. The analysis of mean-field asymptotics provides explicit approximations (often deterministic) for the collective dynamics, fluctuations, and statistical structure of large, interacting systems in physics, biology, engineering, and the mathematical sciences. This article surveys foundational results, methodologies, and implications in the modern paper of the mean-field asymptotic regime, with a focus on statistical inference for interacting diffusions in the mean-field scaling.

1. Mean-Field Regime in Interacting Diffusions

Mean-field models of interacting diffusions are stochastic processes where each particle $X^i_t$ evolves according to dynamics influenced by the empirical distribution of the entire population: $$dX^i_t = b(\vartheta; t, X^i_t, \mu^N_t) dt + \sigma dB^i_t, \qquad \mu^N_t = \frac{1}{N} \sum_{j=1}^N \delta_{X^j_t},$$ where $b$ is a drift depending on a parameter $\vartheta \in \mathbb{R}^p$ and on the empirical measure, $\sigma$ is a diffusion term, and $B^i_t$ are i.i.d. Brownian motions. This "mean-field" scaling encapsulates situations where pairwise interactions become weak as $N$ increases, so the impact of the collective system on any individual becomes deterministic in the limit. As $N \to \infty$, $\mu^N_t$ converges to a non-random measure $\mu^\vartheta_t$ governed by a nonlinear McKean–Vlasov equation.

2. Local Asymptotic Normality (LAN) in Mean-Field Models

A central statistical concept in the mean-field regime is local asymptotic normality (LAN). In this context, LAN asserts that as $N$ grows, the log-likelihood ratio of observing the system under a perturbed parameter $$\vartheta_N = \vartheta + (N\mathbb{I}_\mathcal{G}(\vartheta))^{-1/2}u$$ versus the true parameter $\vartheta$ behaves as: $$\log\frac{dP_{\vartheta_N}^{N}}{dP_{\vartheta}^{N}} = u^\top \xi_N - \frac{1}{2}|u|^2 + r_N(\vartheta,u),$$ where $\xi_N$ converges in law to a standard normal vector and $r_N(\vartheta,u)\to 0$ in probability. The matrix $\mathbb{I}_\mathcal{G}(\vartheta)$ is the asymptotic (limiting) Fisher information for the associated McKean–Vlasov (mean-field) model. This expansion is established using tools such as Girsanov’s theorem, Taylor approximation, propagation-of-chaos, and moment bounds (Maestra et al., 2022). It is critical for establishing the statistical efficiency of estimators.

3. Fisher Information, Identifiability, and Non-Degeneracy

The asymptotic Fisher information $\mathbb{I}_\mathcal{G}(\vartheta)$ quantifies the precision of parameter estimation in the mean-field limit. The normalized Fisher information for the $N$-particle experiment converges: $$N^{-1}\mathbb{I}_{\mathcal{E}^N}(\vartheta) \to \mathbb{I}_\mathcal{G}(\vartheta).$$ Identifiability in this regime is guaranteed if the mapping

$$\vartheta \mapsto \left[(t, x) \mapsto b(\vartheta; t, x, \mu^\vartheta_t)\right]$$

is injective almost everywhere. Non-degeneracy (invertibility) of $\mathbb{I}_\mathcal{G}(\vartheta)$ is certified if, for every direction $z \in \mathbb{R}^p$, the directional derivative $$\nabla_\vartheta (c^{-1/2}b)^j(\vartheta; 0, x, \mu_0)^\top z$$ is not identically zero for at least one $j$. These explicit criteria are essential for determining when sharper asymptotic results apply (Maestra et al., 2022).

4. Optimality and Efficiency of Inference: Hajek's Convolution Theorem

A consequence of the LAN expansion is that the maximum likelihood estimator (MLE) for $\vartheta$ is asymptotically normal and efficient: $$\sqrt{N}\left(\widehat{\vartheta}_N^{\mathrm{MLE}} - \vartheta\right) \xrightarrow{d} \mathcal{N}\left(0, \mathbb{I}_\mathcal{G}(\vartheta)^{-1}\right).$$ Furthermore, Hajek's convolution theorem implies that the MLE achieves the minimal possible asymptotic risk (minimax optimality) among all estimators, up to constants. For any polynomial loss function $w$ and for small perturbations $\theta'$, the minimax risk lower bound is

$$\liminf_{N \to \infty} \sup_{|\theta'-\theta|\leq\delta} E_{P_{\theta'}^N}\left[w\left(\sqrt{N}\, \mathbb{I}_\mathcal{G}(\theta)^{1/2}(\widehat{\theta}_N-\theta')\right)\right] \geq \frac{1}{(2\pi)^{p/2}} \int_{\mathbb{R}^p}w(x)e^{-\frac{1}{2}|x|^2}dx.$$

This lower bound is attained by the MLE, demonstrating minimax optimality for parameter estimation in mean-field interacting diffusions (Maestra et al., 2022).

5. Technical Methodology and Regularity Assumptions

The proof of LAN and accompanying statistical results relies on several technical ingredients:

• Regularity of coefficients: The drift $b(\vartheta; t, x, \nu)$ and its derivatives in $\vartheta$ and $x$ must satisfy uniform smoothness and growth conditions. The diffusion coefficient $\sigma$ must be non-degenerate and regular.
• Propagation of chaos: As $N \rightarrow \infty$, the empirical distribution $\mu^N_t$ converges to the solution $\mu_t$ of the McKean–Vlasov equation. This allows decoupling of particle-level interactions in the limit and reduction to single-particle (mean-field) statistics.
• Coupling arguments: Couplings between the finite-$N$ particle systems and the limiting McKean–Vlasov system are constructed to control the stochastic remainder terms in the LAN expansion.
• Moment controls and inequalities: Detailed control of the likelihood process is needed, using Burkholder–Davis–Gundy, Gronwall, and Rosenthal inequalities to bound error terms arising from the stochastic perturbation and coupling process.

These elements ensure uniformity and tight control needed to justify all limiting operations rigorously (Maestra et al., 2022).

6. Implications for Statistical Inference in Interacting Systems

LAN results in the mean-field regime provide a rigorous foundation for parametric inference in large-scale interacting particle systems. The main implications include:

• Decoupling in the Limit: As $N$ grows, the joint likelihood is close to a product measure of independent McKean–Vlasov diffusions, simplifying statistical analysis.
• Asymptotic Normality: The MLE achieves root-$N$ consistency and asymptotically Gaussian fluctuations with variance determined by the limiting Fisher information.
• Optimal Statistical Procedures: The minimax efficiency ensures that no estimator can outperform the MLE (in risk under polynomial loss) asymptotically.
• Transferable Criteria: Explicit identifiability and non-degeneracy conditions on model parameters can be checked directly in applications to ensure regularity and efficiency.

These results provide both theoretical guarantees and practical tools for inference in models where mean-field interactions play a central role. The rigorous analysis of the mean-field asymptotic regime underpins reliable statistical methodology in fields as diverse as statistical physics, neuroscience, engineering networks, and quantitative biology (Maestra et al., 2022).

7. Summary Table: Key Concepts in Mean-Field LAN Regime

Concept	Description	Mathematical Formulation
Local Asymptotic Normality (LAN)	Log-likelihood ratio approaches a quadratic Gaussian form	$$\log\frac{dP_{\vartheta_N}^{N}}{dP_{\vartheta}^{N}} = u^\top \xi_N - \frac{1}{2}|u|^2 + r_N(\vartheta,u)$$
Fisher Information	Limiting precision for parameter estimation	$$N^{-1}\mathbb{I}_{\mathcal{E}^N}(\vartheta) \to \mathbb{I}_\mathcal{G}(\vartheta)$$
Minimax Optimality	MLE attains infimum of asymptotic risk	$$\liminf_{N \to \infty} \sup_{|\theta'-\theta|\leq\delta} E\ldots \geq (\dots)$$
Identifiability	Unique mapping from parameter to drift	$$\theta \mapsto (t,x)\mapsto b(\theta; t, x, \mu^\theta_t)$$ injective
Non-Degeneracy	Positive-definite Fisher information matrix	$$x \mapsto \nabla_\theta (c^{-1/2}b)^j(\theta;0,x,\mu_0)^\top z$$ not identically zero

This framework, exemplified by the results in the cited work, forms a cornerstone in the rigorous statistical analysis of large-scale interacting stochastic systems in the mean-field regime.