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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.05087 |
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| _version_ | 1866909892419256320 |
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| author | Chen, Yong |
| author_facet | Chen, Yong |
| contents | Using the inner product formula of the canonical Hilbert space of fractional Brownian motion on an interval $[0,T]$ with Hurst parameter $H\in (0,1)$ given by Alazemi et al., we show the asymptotic expansion of the norm of $f_T(s,t):=e^{-|t-s|}\mathbf{1}_{\{0\le s,t\le T\}}$ up to the term $T^{4H-4}$. As applications, we show that the existence of the oblique asymptote of the norm $\frac12\|f_T\|^2_{\mathfrak{H}^{\otimes2}}$ if and only if $H\in (0,\frac12]$ and that we obtain a sharp upper bound of the difference $\left|\frac{1}{2 {T}} \|f_T\|_{\mathfrak{H}^{\otimes 2}}^2-σ^2\right|$ for $H\in (0,\frac34)$ which implies two significant estimates concerning to an ergodic fractional Ornstein-Uhlenbeck process, where $σ^2$ is the slope of the oblique asymptote. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05087 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An asymptotic expansion of the norm of $e^{-|{t-s}|}{1}_{\{0\le s,t\le T\}}$ in the canonical Hilbert space of fractional Brownian motion Chen, Yong Probability 60G22, 41A60, 60H07 Using the inner product formula of the canonical Hilbert space of fractional Brownian motion on an interval $[0,T]$ with Hurst parameter $H\in (0,1)$ given by Alazemi et al., we show the asymptotic expansion of the norm of $f_T(s,t):=e^{-|t-s|}\mathbf{1}_{\{0\le s,t\le T\}}$ up to the term $T^{4H-4}$. As applications, we show that the existence of the oblique asymptote of the norm $\frac12\|f_T\|^2_{\mathfrak{H}^{\otimes2}}$ if and only if $H\in (0,\frac12]$ and that we obtain a sharp upper bound of the difference $\left|\frac{1}{2 {T}} \|f_T\|_{\mathfrak{H}^{\otimes 2}}^2-σ^2\right|$ for $H\in (0,\frac34)$ which implies two significant estimates concerning to an ergodic fractional Ornstein-Uhlenbeck process, where $σ^2$ is the slope of the oblique asymptote. |
| title | An asymptotic expansion of the norm of $e^{-|{t-s}|}{1}_{\{0\le s,t\le T\}}$ in the canonical Hilbert space of fractional Brownian motion |
| topic | Probability 60G22, 41A60, 60H07 |
| url | https://arxiv.org/abs/2511.05087 |