Local Asymptotic Normality (LAN)

• Local Asymptotic Normality (LAN) is a property that approximates the log-likelihood ratio by a Gaussian quadratic form near the true parameter, enabling efficient estimation.
• LAN provides a framework for deriving central sequences and Fisher information matrices, forming the basis for asymptotically optimal tests and estimators.
• The concept is applied in classical, quantum, and nonparametric models, leveraging techniques like Taylor expansion, martingale CLTs, and Girsanov’s theorem.

Local Asymptotic Normality (LAN) is a foundational property in statistical inference, characterizing the asymptotic behavior of likelihood ratios in families of stochastic processes or statistical experiments. Informally, a sequence of statistical models exhibits LAN if, under suitable local reparameterization near a "true" parameter value, the (log-)likelihood ratio between the perturbed and true parameters admits a universal Gaussian quadratic approximation. This property is pivotal in asymptotic decision theory, underlies the construction of efficient estimators, and determines optimal testing strategies. The LAN property has been established—often with considerable technical sophistication—for a broad class of stochastic models, including but not limited to interacting diffusions, jump processes, fractional and mixed Brownian models, nonparametric drift estimation, multi-armed bandits, and quantum statistical models.

1. Definition and General Formalism

Let $(P_\vartheta^n)$ denote a sequence of statistical models, parametrized by $\vartheta$ in an open subset $\Theta$ of $\mathbb{R}^p$ or more generally in a separable Hilbert space. The sequence is said to have the Local Asymptotic Normality (LAN) property at $\vartheta_0$ if there exist:

• A scaling or normalization matrix $\varphi_n(\vartheta_0)$, typically $\varphi_n = n^{-1/2} I_p$ in regular parametric models;
• A random central sequence (or local score) $\Delta_n(\vartheta_0)$;
• A symmetric positive-definite Fisher information matrix $I(\vartheta_0)$;

such that for every fixed $u \in \mathbb{R}^p$,

$\vartheta$0

and under $\vartheta$1, $\vartheta$2. All classical asymptotic theory for efficient estimation (e.g., Cramér–Rao lower bounds, construction of locally most powerful tests, LAN-based minimax theorems) follows from this structure (Maestra et al., 2022, Akker et al., 13 Dec 2025, Ning et al., 2020).

In nonstandard or high-frequency settings the normalization $\vartheta$3 may be non-diagonal or process-dependent, reflecting different estimation rates for different parameters (Brouste et al., 2016, Cai, 30 Dec 2025, Cai, 6 Jan 2026).

2. Methodologies for Establishing LAN

The proof of the LAN property is model-dependent but always relies on a combination of the following elements:

• Taylor Expansion of Log-Likelihood: Expansion in local parameter neighborhoods, isolating linear (score) and quadratic (Fisher information) terms.
• Martingale and Central Limit Theorems: Martingale CLT for the central sequence or its projections is routinely used in models involving stochastic differential equations or Markov processes (Heidari et al., 17 Nov 2025, Liu et al., 2015, Maestra et al., 2022).
• Girsanov/Change-of-Measure Formulas: Girsanov’s theorem for continuous and jump diffusions supplies explicit formulas for likelihood ratios (Maestra et al., 2022, Liu et al., 2015, Kohatsu-Higa et al., 2014).
• Spectral and Semigroup Methods: In nonparametric and time-homogeneous settings (e.g., for reflected diffusions), analytic/spectral PDE tools control the statistical random terms (Wang, 2018).
• Malliavin Calculus & Integration by Parts: In models with jumps or implicitly defined transition densities (e.g., McKean–Vlasov diffusions), Malliavin calculus enables explicit score representations (Heidari et al., 17 Nov 2025, Alaya et al., 2019).
• Operator and Non-diagonal Normalizations: In time-series with strong dependency, e.g., fractional Gaussian noise or mixed fractional Brownian motion, orthogonalizations and rate matrices aligned to the degeneracy/singularity structure are necessary (Cai, 30 Dec 2025, Cai, 6 Jan 2026, Brouste et al., 2016).
• Quantum Statistical Models: LAN extends to finite-dimensional quantum models, using noncommutative Lebesgue decompositions and quantum central limit theory (Fujiwara et al., 2017, Yamagata et al., 2012).

Special care is needed in non-ergodic, singular, or ill-posed statistical models, where the LAN expansion may fail, or the central sequence and Fisher information may be degenerate or require careful re-normalization.

3. LAN in Interacting Particle Systems and McKean-Vlasov SDEs

For mean-field models (McKean–Vlasov SDEs), consider N exchangeable particles: $\vartheta$4 The LAN property for fixed time horizon $\vartheta$5 and high-dimensional $\vartheta$6 is established via a continuous-time Girsanov formula, expanding the log-likelihood for a parameter shift $\vartheta$7: $\vartheta$8 where

$\vartheta$9

(the empirical score) and

$\Theta$0

Key technical requirements include Lipschitz continuity and regularity in $\Theta$1 and $\Theta$2, uniform ellipticity, and injectivity/nondegeneracy of the Fisher information (Maestra et al., 2022). In discrete and high-frequency settings, similar LAN expansions hold but with possibly distinct scaling rates for drift and diffusion (Heidari et al., 17 Nov 2025).

4. LAN in Models with Nonstandard Rates and Degeneracies

Fractional Gaussian Models

In models involving fractional Brownian motion or its increments (fGn), standard diagonal normalization fails due to dependency-induced co-linearity in score vectors. Non-diagonal rate matrices must be constructed:

• For fGn with unknown Hurst parameter $\Theta$3 and scale $\Theta$4:

$\Theta$5

This "rotates" the raw score, yielding asymptotically independent components and non-singular Fisher information (Brouste et al., 2016).

• For mixed fBm models $\Theta$6, further triangular or orthogonalizing transformations are necessary to address degeneracy, especially when $\Theta$7, ensuring the LAN expansion holds with a full-rank (possibly diagonalized) information matrix (Cai, 30 Dec 2025, Cai, 6 Jan 2026).

Multi-Component Parameters and Partial Information

In bandit models and adaptive allocation designs, LAN rates may be componentwise: parameters affecting optimal arms are $\Theta$8-estimable, while those associated solely with suboptimal arms are only estimable at a $\Theta$9-rate, and a block-diagonal information structure arises (Akker et al., 13 Dec 2025).

5. Infinite-Dimensional and Nonparametric LAN

LAN theory generalizes to infinite-dimensional parameter spaces, including nonparametric drift estimation in diffusions and convex M-estimation in Hilbert spaces:

• Nonparametric Drift (Wang, 2018): The score is a directional derivative, and the Fisher information is an operator norm. PDE and spectral methods establish smoothness and rates.
• Convex M-Estimation (Takanashi, 2017): With Mosco-convergence of objective functions (weaker than uniform convergence), LAN holds in the Hilbert space: for $\mathbb{R}^p$0,

$\mathbb{R}^p$1

where $\mathbb{R}^p$2 and $\mathbb{R}^p$3 is a generalized Hessian.

6. Minimax Theory, Efficiency, and Statistical Consequences

The principal consequence of LAN is that the maximum likelihood estimator (MLE), or any regular estimator whose asymptotic behavior matches the LAN quadratic expansion, achieves the minimax lower bound—the so-called convolution bound of Hájek and Le Cam—for local (contiguous) parameter neighborhoods: $\mathbb{R}^p$4 with equality for the asymptotically efficient estimator (Maestra et al., 2022, Cai, 30 Dec 2025, Duarte et al., 5 May 2026). This encompasses the construction of efficient confidence regions, tests, and reduces local inference to the canonical Gaussian shift experiment. Analogous results hold in quantum statistical models, with the optimal estimation rates characterized via the quantum Fisher information and the Holevo bound (Fujiwara et al., 2017, Yamagata et al., 2012).

7. Extensions, Variations, and Related Notions

Several notable generalizations and variants are directly linked to the LAN paradigm:

• Local Asymptotic Quadraticity (LAQ) and Mixed Normality (LAMN): When the Gaussian approximation holds only in a weaker or random-coefficient sense (e.g., supercritical jump CIR, critical bandits), the LAN property is replaced by LAQ or LAMN, and limit likelihood ratios may involve more complex, possibly process-dependent, limit experiments (Alaya et al., 2019).
• Non-linear and Mean-Field Regimes: In interactive particle systems or non-linear McKean–Vlasov models, identifiability and nondegeneracy of the Fisher information may require explicit analytic criteria, often computable via the moments of the limiting empirical measure or propagation-of-chaos principles (Maestra et al., 2022).
• Rescaled LAN (RLAN): For models with growing Monte Carlo error or in large neighborhoods, cubic approximations extend the LAN notion (RLAN), and modified estimators (e.g., maximum cubic likelihood estimator) retain statistical efficiency even as traditional LAN error rates become negligible (Ning et al., 2020).

Table: Representative Models and Features of LAN

Model/Class	Normalization/Rate	Key Techniques
Classical parametric (iid)	$\mathbb{R}^p$5	Score / Fisher info
McKean–Vlasov SDEs (continuous)	$\mathbb{R}^p$6	Girsanov, CLT
High-frequency fGn / mixed fBm	Non-diagonal, componentwise	Orthogonalization
Multi-armed bandits	Block-diagonal, componentwise	Martingale CLT
Nonparametric (diffusion drift)	Local path in Hilbert space	Spectral/PDE methods
Quantum parametric (finite-dim)	$\mathbb{R}^p$7, operator log-like	CCR, SLD, QCLT

LAN is thus the structural backbone of high-dimensional, high-frequency, and non-standard inference for both classical and quantum models, enabling sharp efficiency, testing, and information-theoretic bounds across a wide scope of modern statistical theory (Maestra et al., 2022, Cai, 30 Dec 2025, Brouste et al., 2016, Ning et al., 2020, Heidari et al., 17 Nov 2025, Akker et al., 13 Dec 2025, Fujiwara et al., 2017, Yamagata et al., 2012).