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Local Asymptotic Normality for Mixed Fractional Brownian Motion with $H>3/4$ Under High-Frequency Observation

Chunhao Cai

TL;DR

The paper addresses the local asymptotic normality (LAN) of a mixed fractional Brownian motion with Hurst index $H>3/4$ under high-frequency observations. It develops a two-stage projection of the score, driven by Toeplitz-structured covariance, to overcome a rank-1 degeneracy and yields a diagonal Gaussian LAN expansion with an explicit information matrix $I^{\perp}$. The approach combines exact score representations, CLTs for Gaussian quadratic forms, and matrix Taylor expansions to obtain a robust LAN result and its invariance under reparametrizations. Simulation demonstrates a diagonal projected-score limit and clarifies why projection is essential for nondegenerate inference in this regime.

Abstract

In this paper we will consider the LAN property for both the Hurst parameter $H>3/4$ 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 $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F\to0$. At last we obtain a diagonal Gaussian LAN expansion with an explicit information matrix.

Local Asymptotic Normality for Mixed Fractional Brownian Motion with $H>3/4$ Under High-Frequency Observation

TL;DR

The paper addresses the local asymptotic normality (LAN) of a mixed fractional Brownian motion with Hurst index under high-frequency observations. It develops a two-stage projection of the score, driven by Toeplitz-structured covariance, to overcome a rank-1 degeneracy and yields a diagonal Gaussian LAN expansion with an explicit information matrix . The approach combines exact score representations, CLTs for Gaussian quadratic forms, and matrix Taylor expansions to obtain a robust LAN result and its invariance under reparametrizations. Simulation demonstrates a diagonal projected-score limit and clarifies why projection is essential for nondegenerate inference in this regime.

Abstract

In this paper we will consider the LAN property for both the Hurst parameter 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 -score linear term and orthogonalize the remainder through two non-diagonal transformations, then we can construct the CLT for the quadratic form base on . At last we obtain a diagonal Gaussian LAN expansion with an explicit information matrix.
Paper Structure (24 sections, 22 theorems, 192 equations, 3 figures)

This paper contains 24 sections, 22 theorems, 192 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Exact score functions
  4. Preliminaries: Score Functions
  5. Some General Properties of fGn and Score Function
  6. Central Limit Theorem for $S_{\sigma,n}$
  7. The CLT for the Remainder of Score function of $H$
  8. CLT for the projected score vector
  9. Projection and exact orthogonality
  10. Asymptotics of the projection coefficient and the orthogonal Frobenius norm
  11. Joint CLT after multiplying by the rate matrix
  12. LAN Property---Proof of Theorem \ref{['thm:LAN-rank2']}
  13. Matrix Taylor Expansions
  14. Final Proof of Theorem \ref{['thm:LAN-rank2']}
  15. Simulation Study
  16. ...and 9 more sections

Key Result

Theorem 1

Assume $\bigl(\mathbf P_\theta^n\bigr)_{\theta\in\Theta}$ is LAN at $\theta_0$ with normalization matrices $\phi(n)$ and information $I(\theta_0)$. Then for any sequence of estimators $\hat{\theta}_n$, any $\ell\in W_{2,k}$, and any $\delta>0$, where $\varphi_{I(\theta_0)^{-1}}$ denotes the centered Gaussian density with covariance $I(\theta_0)^{-1}$ (restricted to the range of $I(\theta_0)$ when

Figures (3)

  • Figure 1: Projected/orthogonalized score $\Xi_n$ (simulation) with theoretical Gaussian contours.
  • Figure 2: Limiting Gaussian surface density of the projected score.
  • Figure 3: Without projection: singular (rank-$1$) limit; the normalized pair concentrates near a line.

Theorems & Definitions (49)

  • Definition 1: LAN
  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 39 more