Local Asymptotic Normality for Mixed Fractional Brownian Motion Under High-Frequency Observation

TL;DR

This paper establishes the local asymptotic normality (LAN) property for mixed fractional Brownian motion with high-frequency data, providing explicit Gaussian expansions and demonstrating the method's applicability across different Hurst parameters.

Contribution

The paper introduces a novel approach using non-diagonal transformations and quadratic form CLT to prove LAN for mixed fractional Brownian motion, extending analysis to both H>3/4 and H<3/4 cases.

Findings

Derived explicit Gaussian LAN expansion with information matrix.

Proved the method's effectiveness for H<3/4 case.

Applicable to fractional Ornstein-Uhlenbeck models.

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. Beyond the case of $H>3/4$, we also present that the $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F\to0$ method is also useful for the case of $H<3/4$ 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.