[Paper Review] Local Asymptotic Normality for Mixed Fractional Brownian Motion Under High-Frequency Observation

Chunhao Cai, Yiwu Shang|arXiv (Cornell University)|Jan 6, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper establishes local asymptotic normality (LAN) for a mixed fractional Brownian motion observed at high frequency, by orthogonalizing the score and deriving a diagonal Gaussian LAN expansion with an explicit information matrix; it also discusses applicability to H<3/4.

ABSTRACT

In this paper we will consider the LAN property for both the Hurst parameter $H&gt;3/4$ and the variance of the fractional Brownian motion plus an independent standard Brownian motion (called mixed fractional Brownian motion) with high-frequency observation. We will first remove the $H$-score linear term and orthogonalize the remainder through two non-diagonal transformations, then we can construct the CLT for the quadratic form base on $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F o0$. At last we obtain a diagonal Gaussian LAN expansion with an explicit information matrix. Beyond the case of $H&gt;3/4$, we also present that the $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F o0$ method is also useful for the case of $H&lt;3/4$ and the proof will be concise compared with the Whittle translation method. We consider that this method can be applied to this type of problem, including the fractional Ornstein-Uhlenbeck model and mixed fractional O-U process.

Motivation & Objective

• Motivate statistical inference for a mixed fractional Brownian motion (mfBm) model with unknown volatility and Hurst parameter.
• Develop LAN property under a high-frequency infill scheme for H in (3/4,1).
• Provide a constructive path to an explicit Gaussian LAN expansion with a diagonal information matrix.
• Explain why the method extends beyond H>3/4 and compare with Whittle-type approaches.

Proposed method

• Define mfBm Y_t = σ B^H_t + B_t with unknown θ=(σ,H) and high-frequency discrete observations.
• Express the exact Gaussian log-likelihood and derive score components S_{σ,n} and S_{H,n} as centered quadratic forms.
• Perform a two-step non-diagonal transformation to orthogonalize the score and obtain a diagonal LAN form.
• Use Toeplitz covariance structure and Fisher–Hartwig-type representations to derive trace approximations for regularized Toeplitz matrices.
• Develop CLTs for the score components via a norm-compatibility condition ||M_n||_op / ||M_n||_F → 0 and a quadratic-form CLT (Lemma 2).
• Compute the asymptotic Fisher information matrix I^⊥ with blocks involving J_0, J_⊥, and provide explicit representations (via J_0,J_1,J_2).

Experimental results

Research questions

• RQ1 Can LAN be established for mfBm under high-frequency observation when H>3/4, and what is the structure of the limiting information?
• RQ2 How to overcome score degeneracy due to near-zero frequency dominance by orthogonalization?
• RQ3 Does the same approach extend to H<3/4, and how does it compare to Whittle-type methods?
• RQ4 What are the explicit forms and roles of the asymptotic information components J_0, J_1, J_2 in the LAN expansion?

Key findings

• The statistical model with X_n ~ N(0,V_n(θ)) admits an LAN expansion with r_n scaling and a diagonal Gaussian limit for θ=(σ,H).
• The score vector becomes degenerate for H>3/4 without orthogonalization, which is resolved by two non-diagonal transformations.
• The asymptotic information matrix I^⊥ is diagonal and partitioned, with blocks involving J_0(H,σ) and J_⊥(H,σ) (where J_⊥ = J_2 − J_1^2/J_0).
• Explicit trace expansions for quadratic forms with Toeplitz matrices are derived, enabling CLTs for S_{σ,n} and the H-remainder.
• A generalized Szegő-type trace approximation for triangular arrays under regularization is established to handle the n-dependent spectral behavior.
• The method is suggested to extend to the fractional Ornstein–Uhlenbeck model and mixed fractional OU processes.