Lower classes and Chung's LILs of the fractional integrated generalized fractional Brownian motion
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arXiv
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| Format: | Preprint |
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2024
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| author | Lyu, Mengjie Wang, Min Wang, Ran |
| author_facet | Lyu, Mengjie Wang, Min Wang, Ran |
| contents | Let $\{X(t)\}_{t\geqslant0}$ be the generalized fractional Brownian motion introduced by Pang and Taqqu (2019):
\begin{align*}
\{X(t)\}_{t\ge0}\overset{d}{=}&\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ/2}
B(du) \right\}_{t\ge0},
\end{align*} where $ γ\in [0,1), \ \ α\in \left(-\frac12+\fracγ{2}, \ \frac12+\fracγ{2} \right)$ are constants. For any $θ>0$, let \begin{align*}
Y(t)=\frac{1}{Γ(θ)}\int_0^t (t-u)^{θ-1} X(u)du, \quad t\ge 0. \end{align*} Building upon the arguments of Talagrand (1996), we give integral criteria for the lower classes of $Y$ at $t=0$ and at infinity, respectively. As a consequence, we derive its Chung-type laws of the iterated logarithm. In the proofs, the small ball probability estimates play important roles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11851 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower classes and Chung's LILs of the fractional integrated generalized fractional Brownian motion Lyu, Mengjie Wang, Min Wang, Ran Probability 60G15, 60G17, 60G18, 60G22 Let $\{X(t)\}_{t\geqslant0}$ be the generalized fractional Brownian motion introduced by Pang and Taqqu (2019): \begin{align*} \{X(t)\}_{t\ge0}\overset{d}{=}&\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ/2} B(du) \right\}_{t\ge0}, \end{align*} where $ γ\in [0,1), \ \ α\in \left(-\frac12+\fracγ{2}, \ \frac12+\fracγ{2} \right)$ are constants. For any $θ>0$, let \begin{align*} Y(t)=\frac{1}{Γ(θ)}\int_0^t (t-u)^{θ-1} X(u)du, \quad t\ge 0. \end{align*} Building upon the arguments of Talagrand (1996), we give integral criteria for the lower classes of $Y$ at $t=0$ and at infinity, respectively. As a consequence, we derive its Chung-type laws of the iterated logarithm. In the proofs, the small ball probability estimates play important roles. |
| title | Lower classes and Chung's LILs of the fractional integrated generalized fractional Brownian motion |
| topic | Probability 60G15, 60G17, 60G18, 60G22 |
| url | https://arxiv.org/abs/2405.11851 |