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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.02482 |
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| _version_ | 1866908603270561792 |
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| author | Tang, Zheng Li, Ying Yang, Haili Yi, Hua Chen, Yong |
| author_facet | Tang, Zheng Li, Ying Yang, Haili Yi, Hua Chen, Yong |
| contents | Let the Ornstein-Uhlenbeck process $\{X_t,\,t\geq 0\}$ driven by a fractional Brownian motion $B^H$ described by $d X_t=-θX_t dt+ d B_t^H,\, X_0=0$ with known parameter $H\in (0,\frac34)$ be observed at discrete time instants $t_k=kh, k=1,2,\dots, n $. If $θ>0$ and if the step size $h>0$ is arbitrarily fixed, we derive Berry-Esséen bound for the ergodic type estimator (or say the moment estimator) $\hatθ_n$, i.e., the Kolmogorov distance between the distribution of $\sqrt{n}(\hatθ_n-θ)$ and its limit distribution is bounded by a constant $C_{θ, H,h}$ times $n^{-\frac12}$ and $ n^{4H-3}$ when $H\in (0,\,\frac58]$ and $H\in (\frac58,\,\frac34)$, respectively. This result greatly improve the previous result in literature where $h$ is forced to go zero. Moreover, we extend the Berry-Esseen bound to the Ornstein-Uhlenbeck model driven by a lot of Gaussian noises such as the sub-bifractional Brownian motion and others. A few ideas of the present paper come from Haress and Hu (2021), Sottinen and Viitasaari (2018), and Chen and Zhou (2021). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02482 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Berry-Esseen bound for the Moment Estimation of the fractional Ornstein-Uhlenbeck model under fixed step size discrete observations Tang, Zheng Li, Ying Yang, Haili Yi, Hua Chen, Yong Probability Let the Ornstein-Uhlenbeck process $\{X_t,\,t\geq 0\}$ driven by a fractional Brownian motion $B^H$ described by $d X_t=-θX_t dt+ d B_t^H,\, X_0=0$ with known parameter $H\in (0,\frac34)$ be observed at discrete time instants $t_k=kh, k=1,2,\dots, n $. If $θ>0$ and if the step size $h>0$ is arbitrarily fixed, we derive Berry-Esséen bound for the ergodic type estimator (or say the moment estimator) $\hatθ_n$, i.e., the Kolmogorov distance between the distribution of $\sqrt{n}(\hatθ_n-θ)$ and its limit distribution is bounded by a constant $C_{θ, H,h}$ times $n^{-\frac12}$ and $ n^{4H-3}$ when $H\in (0,\,\frac58]$ and $H\in (\frac58,\,\frac34)$, respectively. This result greatly improve the previous result in literature where $h$ is forced to go zero. Moreover, we extend the Berry-Esseen bound to the Ornstein-Uhlenbeck model driven by a lot of Gaussian noises such as the sub-bifractional Brownian motion and others. A few ideas of the present paper come from Haress and Hu (2021), Sottinen and Viitasaari (2018), and Chen and Zhou (2021). |
| title | Berry-Esseen bound for the Moment Estimation of the fractional Ornstein-Uhlenbeck model under fixed step size discrete observations |
| topic | Probability |
| url | https://arxiv.org/abs/2504.02482 |