LAN Property in Gaussian Processes

• LAN Property is a statistical concept where local parameter perturbations yield a Gaussian shift approximation of the log-likelihood ratio, establishing a basis for optimal inference.
• It employs detailed mathematical tools such as Toeplitz matrices and spectral density analysis to address challenges in long-memory and fractional-type models.
• The property underpins the efficiency of maximum likelihood estimators and extends classical inference methods to models with non-standard noise and long-range dependence.

The LAN (Local Asymptotic Normality) property is a central asymptotic concept in modern statistics governing the local behavior of statistical experiments under parameter perturbations. In its canonical form, LAN formalizes the convergence of the log-likelihood ratio to a Gaussian shift experiment, providing the foundation for optimal parametric inference. The mathematical structure, general conditions, and implications in various models, especially those with long-range dependence or non-standard noise, have been the subject of rigorous analysis. This article comprehensively covers the definition, mathematical framework, sufficient conditions, main theoretical results, and applications of the LAN property, with a focus on Gaussian processes with fractional-type or long-memory dependence (Cohen et al., 2011).

1. Definition and Statistical Significance of the LAN Property

The LAN property expresses that the statistical experiment locally, around a true parameter $\theta_0$, mimics a Gaussian shift experiment under an appropriate normalization. For a family of distributions $(P^n_\theta)$ indexed by a parameter $\theta \in \Theta \subset \mathbb{R}^d$, the model is said to possess the LAN property at $\theta_0$ if, for any sequence of local perturbations $\theta_n = \theta_0 + t/\sqrt{n}$ with $t \in \mathbb{R}^d$ fixed, the log-likelihood ratio admits the expansion: $$\log \frac{dP^n_{\theta_n}}{dP^n_{\theta_0}}(X) = \langle t, Z_n \rangle - \frac{1}{2} t^* I(\theta_0) t + \psi_{\theta_0}(t, n)$$ where:

• $Z_n$ is a random vector (the normalized score) converging in distribution to a centered Gaussian with covariance $I(\theta_0)$, the Fisher information matrix;
• $\psi_{\theta_0}(t, n)$ is a remainder term converging to $0$ uniformly on compact sets in $t$;
• $I(\theta_0)$ characterizes the local information geometry.

The LAN property is fundamental because it enables rigorous proofs of asymptotic normality, efficiency, and minimaxity for maximum likelihood and regular estimators in locally quadratic log-likelihood models. Results such as the Hájek–Le Cam convolution and minimax theorems become available, justifying the use of Fisher information-based asymptotics and allowing transfer of optimality statements from the Gaussian shift model to more complex or dependent data models.

2. Mathematical Setting: Gaussian Processes, Spectral Density, and Toeplitz Structures

The fundamental setting of (Cohen et al., 2011) is that of stationary centered Gaussian processes $(X_n)$ whose law is determined by a spectral density $f_\theta(x)$ depending smoothly on a (possibly multidimensional) parameter $\theta$. The main assumption is that, at low frequencies, the spectral density exhibits power-law singularity of the form: $$f_\theta(x) \sim_{x\to 0} |x|^{-\alpha(\theta)} L_\theta(x)$$ where:

• $\alpha(\theta) \in (-\infty, 1)$: determines the memory regime (short, long, or antipersistent),
• $L_\theta(x)$: a slowly varying function at $x=0$ (i.e., for any $t>0$, $L_\theta(tx)/L_\theta(x) \to 1$ as $x \to 0$).

Processes characterized by such spectra include fractional Brownian motion and ARFIMA models, where the dependence structure is governed by the exponent $\alpha(\theta)$. The observed covariance matrix for finite samples $(X_1, \ldots, X_n)$ is a Toeplitz matrix

$$T_n(f) = \bigl[ \int_{-\pi}^\pi e^{i(k-j)x} f(x) dx \bigr]_{1 \leq k, j \leq n}$$

whose spectral asymptotics control the likelihood expansion.

The technical analysis requires:

• Uniform smoothness (typically up to third derivatives in $\theta$) of $x \mapsto f_\theta(x)$,
• Spectral estimates for $T_n(f_\theta)$ and its derivatives,
• Uniform trace and norm bounds for combinations of inverse Toeplitz matrices and their products with differentiated spectral densities,
• Quantitative versions of strong Szegő and Grenander–Szegő theorems for non-smooth (power-law) densities.

These tools are necessary to realize a Taylor expansion of the log-likelihood and control error terms precisely in the presence of non-standard memory and regularity.

3. Main Result: LAN Expansion for Fractional-Type Models

The central technical result established in (Cohen et al., 2011) is that, subject to suitable regularity and slow variation conditions on $f_\theta$ (see Assumptions (A.1)-(A.2)), the LAN property holds for sequences of observations from such Gaussian processes. Specifically,

$$\log \left( \frac{dP^n_{\theta_0 + t/\sqrt{n}}}{dP^n_{\theta_0}}(X) \right) = \langle t, Z_n \rangle - \frac{1}{2} t^* I(\theta_0) t + \psi_{\theta_0}(t, n)$$

where:

• $Z_n$ is a sum of normalized derivatives of the log-likelihood, converging to $N(0, I(\theta_0))$,
• $I(\theta_0)$ admits expressions in terms of spectral integrals involving first and second derivatives of $f_\theta$,
• The error $\psi_{\theta_0}(t, n) \to 0$ uniformly as $n \to \infty$.

Explicit uniform trace and norm bounds are established for quantities of type

$$\operatorname{tr} \Bigl [ T_n(f_\theta)^{-1} T_n\Bigl(\frac{\partial f_\theta}{\partial \theta_\ell}\Bigr) \Bigr ]$$

which are crucial for controlling second and third-order Taylor expansion terms in the log-likelihood. The methodology extends beyond stationary models and accommodates discretized increments of fractional Brownian motion, with further technical handling of nonstationarity via approximation by stationary increments.

Even models at or near the boundary of the long-memory regime ($\alpha(\theta) \approx 1$) are treated, provided uniformity in the error terms and estimates is maintained using properties of Toeplitz matrices indexed by singular or strongly varying spectral densities.

4. Applications, Efficiency, and Broader Impact

Establishing the LAN property under these general assumptions has several immediate mathematical and statistical consequences:

• Efficiency of MLE and Regular Estimators: Asymptotic normality and efficiency of the maximum likelihood estimator follow directly, with limiting variance inversely proportional to $I(\theta_0)$. No loss arises even in the presence of long-range dependence or spectral singularities.
• Extension to Nonstationary and Long-Memory Settings: The analysis seamlessly includes (i) stationary increments of fractional processes and (ii) ARFIMA models, which exhibit nonstandard dependence and may lack traditional asymptotic independence among distant observations.
• Semiparametric and Nonparametric Inference: The uniform LAN result provides the critical technical foundation for higher-level statements, such as semiparametric efficiency bounds and adaptive inference, in models not satisfying short-memory conditions.
• Practical Estimation: LAN implies that, under increasing sample size, complex time series models (for instance, with slowly varying volatility or spectral shape) may be reduced locally to well-characterized Gaussian shift experiments—facilitating the use of classical estimation and testing theory.
• Scientific Domains: Models exhibiting the fractional-type property appear in econometrics, telecommunications, climatology, hydrology, and the physical sciences, among others, where long memory or antipersistent processes are empirically observed. The work justifies the use of Gaussian and likelihood-based asymptotics in these fields, even in unconventional or irregular regimes.

5. Technical Conditions and Examples

The general regularity hypothesis for the main theorems includes:

• $f_\theta(x)$ is three times continuously differentiable in $\theta$ for almost every $x$,
• Uniform bounds: $$c_1 |x|^{-\alpha(\theta) + \dots} \leq f_\theta(x) \leq c_2 |x|^{-\alpha(\theta) - \dots}$$ for some $c_1, c_2 > 0$, and similar bounds on $\partial f_\theta/\partial x$,
• Slow variation: $L_\theta(x)$ is slowly varying as $x \to 0$ with suitable modulus of continuity,
• Smoothness of parameter dependence in both exponent $\alpha(\theta)$ and slow variation $L_\theta$.

Examples covered under this framework:

• Fractional Brownian motion increments (fractional Gaussian noise) with $H > 1/2$,
• ARFIMA$(p, d, q)$ processes for $d < 1/2$,
• Long-memory models with spectral density $f_\theta(x) \sim |x|^{-2d} L_\theta(x)$ for small $x$,
• Models displaying antipersistence, $\alpha(\theta) < 0$, including relevant negative memory scenarios.

6. Unified View and Further Directions

By giving a detailed analysis of the log-likelihood expansion around the true parameter for a broad class of Gaussian processes parametrized by irregular or slowly-varying spectral densities, the LAN property is established as a unifying principle. This includes not only stationary models but also a variety of non-standard stochastic processes encountered in modern applied statistics.

The work paves the way for further developments:

• Refinement of numerical techniques for Fisher information estimation in singular or long-memory regimes,
• Quantitative analysis of nonparametric likelihood behaviors using LAN approximations,
• Design and analysis of modern statistical methods (e.g., Bayesian procedures, penalty-based estimators) under long-range dependency,
• Extension to high-dimensional or functional parameter settings, where analogous spectral and Toeplitz techniques may be adapted.

Summary Table: Key Ingredients in the Proof and Application of LAN

Component	Structure / Condition	Role in Theory
Spectral Density $f_\theta$	$|x|^{-\alpha(\theta)}L_\theta(x)$ near $x=0$	Controls memory regime & dependence
Toeplitz Matrix $T_n(f_\theta)$	$n \times n$ matrix built from $f_\theta$	Encodes sample covariance structure
Regularity/Smoothness	3× differentiability in $\theta$; slow variation in $x$	Allows Taylor expansion; uniform bounds
Fisher Information $I(\theta_0)$	Integral expression in $f_\theta$ and its derivatives	Asymptotic variance of estimators
Uniform Asymptotics	All technical results hold uniformly in $\theta$	Enables robust inference

Conclusion

Local asymptotic normality for Gaussian processes with spectral density of the type $f_\theta(x)\sim|x|^{-\alpha(\theta)} L_\theta(x)$ as $x\to 0$ provides a rigorous and broadly applicable framework for inferential optimality and normal approximation in a spectrum of long-memory and fractional models. The results (Cohen et al., 2011) unify multiple classes of such processes and supply the technical underpinnings for both classical (maximum likelihood) and modern semiparametric and nonparametric approaches to statistical inference in this context.