Local Asymptotic Normality for Mixed Fractional Brownian Motion Under High-Frequency Observation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866909995175510016 |
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| author | Cai, Chunhao Shang, Yiwu |
| author_facet | Cai, Chunhao Shang, Yiwu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02622 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Asymptotic Normality for Mixed Fractional Brownian Motion Under High-Frequency Observation Cai, Chunhao Shang, Yiwu Probability 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. |
| title | Local Asymptotic Normality for Mixed Fractional Brownian Motion Under High-Frequency Observation |
| topic | Probability |
| url | https://arxiv.org/abs/2601.02622 |