Ibragimov–Khasminskii LAN Framework

• Ibragimov–Khasminskii LAN framework is a rigorous statistical theory providing local expansions of log-likelihood ratios to analyze complex, dependent, and singular models.
• It employs advanced tools such as Toeplitz matrix asymptotics, Malliavin calculus, and martingale decompositions to handle non-i.i.d. and jump processes effectively.
• The framework underpins asymptotic normality, efficiency of estimators, and sharp minimax risk bounds across applications like time series, diffusions, and mixed models.

The Ibragimov–Khasminskii LAN (Local Asymptotic Normality) framework is a fundamental statistical theory for the asymptotic analysis of parametric statistical models, particularly in non-i.i.d., dependent, or singular settings. Initially introduced by Ibragimov and Khasminskii as an extension of Le Cam’s LAN, it provides rigorous asymptotic expansions for likelihood ratios, enabling the study of efficiency, minimaxity, and limit experiments in a broad spectrum of stochastic processes—ranging from long-memory Gaussian sequences and ergodic diffusions to models with jumps, fractional structures, and partially observed/mixed systems.

1. Foundations and Definitions

The core concept of the Ibragimov–Khasminskii LAN framework is the local expansion of the log-likelihood ratio between neighboring parameter values, typically in the form: $$\log \frac{dP^n_{\theta+h/\sqrt{n}}}{dP^n_\theta}(X^{(n)}) = \langle h, \Delta_n(\theta)\rangle - \frac{1}{2}h^\top I(\theta) h + o_{P_\theta}(1)$$ where $P^n_\theta$ is the law of the observed process under parameter $\theta$, $h$ is a local perturbation, $\Delta_n(\theta)$ is the central sequence (score), and $I(\theta)$ the Fisher information matrix. The precise definition and scaling depend on the model structure and, for example, in time series or models with long memory, the scaling may be different from $\sqrt{n}$ to account for dependence or singularities in the spectral density (Cohen et al., 2011).

A statistical model is said to be LAN at $\theta_0$ if: (i) the log-likelihood ratio is locally quadratic, (ii) the central sequence is asymptotically Gaussian with mean zero and the same variance as the Fisher information, and (iii) certain regularity (such as contiguity and quadratic mean differentiability) conditions are met.

2. Key Analytical Techniques

The realization of the Ibragimov–Khasminskii LAN property in concrete models involves a combination of advanced probabilistic and functional analytic tools:

• Toeplitz matrix asymptotics are used for stationary Gaussian sequences with spectral densities of the form $f_\theta(x)\sim |x|^{-\alpha(\theta)}L_\theta(x)$, allowing precise control of quadratic forms and traces essential for deriving central sequences and Fisher information (Cohen et al., 2011).
• Malliavin calculus and Girsanov’s theorem are applied in models with Brownian or jump-driven noise to express the score as a martingale or Skorohod integral (e.g., for ergodic diffusions with jumps, Ornstein–Uhlenbeck with Poisson noise, and McKean–Vlasov mean-field models) (Kohatsu-Higa et al., 2015, Tran, 2015, Kohatsu-Higa et al., 2014, Maestra et al., 2022).
• Limit theorems for functionals of Markov chains or path-segments are crucial in time-inhomogeneous or periodic models, often via the ergodicity of “grid chains” or segment chains (Hoepfner et al., 2010, Hoepfner et al., 2010).
• Bracketing and non-asymptotic analysis: Explicit local quadratic bracketing inequalities for finite-sample (non-asymptotic) log-likelihood behavior extend the classical LAN approach to finite $n$ and possible misspecification (Spokoiny, 2011).

This machinery ensures that the key technical requirements—control and normalization of martingale terms, tightness/negligibility of remainders, and appropriate scaling—are satisfied across a diverse array of models.

3. Canonical Model Classes and Typical Expansions

The framework covers a broad class of parametric models, each with a form of the LAN property:

Model Class	Scaling of Parameter Shift	Key Central Sequence
Gaussian long-memory	$h/\sqrt{n}$	Toeplitz quadratic form
Periodic diffusions	$h/\sqrt{n}$ (smooth), $h/n^{2}$ (discont.)	Path-segment martingales
Diffusions with jumps	$h/\sqrt{n\Delta_n}$	Martingale array + Jump terms
Ornstein–Uhlenbeck w/ jumps	$$\operatorname{diag}(\sqrt{n\Delta_n},\sqrt{n},\sqrt{n\Delta_n})$$	Array of conditional score terms
McKean–Vlasov	$h/\sqrt{N}$	Sum of stochastic integrals over particles
Partially observed mixed-fBs	$h/\sqrt{T}$	Martingale in innovation process

For fractional Gaussian processes and ARFIMA models, LAN holds with explicit central sequences involving Toeplitz operators, and Fisher information is computed as an integral over the spectral domain (Cohen et al., 2011). In jump-diffusions or jump-driven OU processes, the log-likelihood ratio is decomposed into sums of increments on small intervals, with explicit score vectors for drift, diffusion, and jump intensity (Kohatsu-Higa et al., 2015, Kohatsu-Higa et al., 2014, Tran, 2015).

In time-inhomogeneous diffusion models with periodicity or discontinuity, the nature of the limiting experiment depends critically on the regularity of the signal; smooth regimes yield classic LAN and efficient estimators, while signals with jumps fall into the “discontinuous signal” Ibragimov–Khasminskii canonical experiment, with only $\frac{1}{2}$–Hölder continuity of the Hellinger distance and a fundamentally non-Gaussian limit (Hoepfner et al., 2010).

4. Main Theorems and Statistical Implications

Once the LAN expansion is verified, the classical statistical implications from Le Cam and Ibragimov–Khasminskii theory are unlocked:

• Asymptotic normality and efficiency: The maximum likelihood estimator (MLE) (or plug-in/one-step corrected estimators) is asymptotically normal with covariance given by the inverse Fisher information. This holds for a range of models, including time series, diffusions with jumps, and partially observed/mixed systems (Cohen et al., 2011, Hoepfner et al., 2010, Cai, 17 Dec 2025, Maestra et al., 2022).
• Minimax bounds: The minimax lower bound for risk under polynomial loss is attained, up to explicit constants, by the MLE and matches the convolution theorem’s bound in the relevant Gaussian or canonical limit experiment (Maestra et al., 2022, Hoepfner et al., 2010).
• Finite-sample accuracy and confidence sets: Non-asymptotic results, such as those of Spokoiny’s local quadratic bracketing, precisely quantify the radius of confidence regions and deviation of the estimator for any fixed sample size (Spokoiny, 2011).
• Irregular cases: In “change-point” or discontinuous signal models, the rate of convergence drops (e.g., $n^{-2}$), and the limit experiment involves functionals of double-sided Brownian motion with linear drift, leading to optimality properties in the sense of the Ibragimov–Khasminskii canonical discontinuous experiment (Hoepfner et al., 2010).

5. Methodological Distinctions and Generalizations

The Ibragimov–Khasminskii LAN framework generalizes Le Cam’s approach in several important directions:

• Non-i.i.d. and time series settings: By leveraging Toeplitz theory and ergodicity of segment-chains, it applies naturally to long-memory, dependent, and nonstationary models.
• Irregular and singular models: The theory is robust to models lacking quadratic mean differentiability (e.g., those with discontinuities, long memory, or non-invertible Fisher information over subspaces).
• Jump and hybrid processes: Incorporation of jump processes or other non-Gaussian noise is handled via Malliavin calculus and advanced probabilistic techniques, expanding applicability to Ornstein–Uhlenbeck with Poisson noise, multidimensional jump-diffusions, and particle systems.
• Non-asymptotic and high-dimensional regimes: The quadratic bracketing approach provides accuracy guarantees and risk bounds in finite sample and broad dimensional regimes (Spokoiny, 2011).
• Partial observation and filtering: The framework adapts to partially observed dynamics (e.g., mixed fractional Ornstein–Uhlenbeck observed through noisy channels) with innovation representations and associated Kalman–Bucy filtering/Riccati equations, where the LAN expansion is based on the innovation process (Cai, 17 Dec 2025).

6. Technical Innovations and Proof Strategies

Several technical advances underpin realizations of the framework across different models:

• Spectral and Toeplitz analysis ensures that the covariance structure of long-memory Gaussian sequences satisfies LAN conditions through operator norm and trace asymptotics (Cohen et al., 2011).
• Triangular array CLT and martingale decompositions allow reductions of compressed high-frequency data (with or without jumps) to Gaussian limit experiments via homogenization and Lindeberg-type arguments (Kohatsu-Higa et al., 2014, Tran, 2015).
• Malliavin calculus provides tractable representations of score functions and their moment properties in jump-diffusion models (Kohatsu-Higa et al., 2015).
• Propagation-of-chaos and mean-field limit techniques identify the limiting Fisher information in interacting particle systems, accommodating nonlinear dependence on empirical measures (Maestra et al., 2022).
• Quadratic bracketing and deviation inequalities extend the essential LAN implications to finite-sample and even misspecified-model settings (Spokoiny, 2011).

7. Applications and Broader Impact

The Ibragimov–Khasminskii LAN framework is central to the modern asymptotic theory of parametric inference in dependent, nonregular, or infinite-dimensional settings. Its consequences include:

• Construction of efficient estimators and asymptotically optimal tests in Gaussian and non-Gaussian time series, diffusions (with or without jumps), and fractional or mixed models (Cohen et al., 2011, Kohatsu-Higa et al., 2015, Cai, 17 Dec 2025, Maestra et al., 2022).
• Sharp minimax lower bounds and efficiency characterizations, even in regimes with non-i.i.d. dependence, fractional characteristics, or partial observation (Hoepfner et al., 2010, Spokoiny, 2011).
• Critical insights into the nature of limit experiments in irregular or singular settings, such as discontinuous signals, with implications for optimal Bayesian and frequentist procedures (Hoepfner et al., 2010).
• Extension of the efficiency theory to parametric inference for McKean–Vlasov systems and high-dimensional or misspecified models (Maestra et al., 2022, Spokoiny, 2011).

Through these developments, the framework provides a rigorous and unifying analytic basis for advanced parametric inference in a wide range of stochastic models relevant to applied probability, time series analysis, filtering, systems biology, and mean-field statistical mechanics.

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References

• "LAN property for some fractional type Brownian motion" (Cohen et al., 2011)
• "On LAN for parametrized continuous periodic signals in a time inhomogeneous diffusion" (Hoepfner et al., 2010)
• "Estimating a periodicity parameter in the drift of a time inhomogeneous diffusion" (Hoepfner et al., 2010)
• "LAN property for an ergodic diffusion with jumps" (Kohatsu-Higa et al., 2015)
• "LAN property for a linear model with jumps" (Kohatsu-Higa et al., 2014)
• "LAN property for an ergodic Ornstein-Uhlenbeck process with Poisson jumps" (Tran, 2015)
• "Drift estimation for a partially observed mixed fractional Ornstein–Uhlenbeck process" (Cai, 17 Dec 2025)
• "The LAN property for McKean-Vlasov models in a mean-field regime" (Maestra et al., 2022)
• "Parametric estimation. Finite sample theory" (Spokoiny, 2011)