Strassen's local law of the iterated logarithm for the generalized fractional Brownian motion
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914580845821952 |
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| author | Wang, Ran Xiao, Yimin |
| author_facet | Wang, Ran Xiao, Yimin |
| contents | Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion given by $$
\{X(t)\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ/2} B(du) \right\}_{t\ge0}, $$ with parameters $γ\in (0, 1)$ and $α\in \left(-1/2+ γ/2, \, 1/2+γ/2\right)$. This process was introduced by Pang and Taqqu (2019) as the scaling limit of a class of power-law shot noise processes. The parameters $α$ and $γ$ govern the probabilistic and statistical properties of $X$. In particular, the parameter $γ$ breaks the stationarity of increments of $X$.
In this paper, we establish Strassen's local law of the iterated logarithm for $X$ at a given point $t_0 \in (0, \infty)$. This result describes explicitly the roles played by the parameters $α, γ$, and the location $t_0$. Our theorem differs from the earlier Strassen's {global law of the iterated logarithm} for $X$ proved by Ichiba, Pang and Taqqu (2022). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strassen's local law of the iterated logarithm for the generalized fractional Brownian motion Wang, Ran Xiao, Yimin Probability 60G15, 60G17, 60G18, 60G22 Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion given by $$ \{X(t)\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ/2} B(du) \right\}_{t\ge0}, $$ with parameters $γ\in (0, 1)$ and $α\in \left(-1/2+ γ/2, \, 1/2+γ/2\right)$. This process was introduced by Pang and Taqqu (2019) as the scaling limit of a class of power-law shot noise processes. The parameters $α$ and $γ$ govern the probabilistic and statistical properties of $X$. In particular, the parameter $γ$ breaks the stationarity of increments of $X$. In this paper, we establish Strassen's local law of the iterated logarithm for $X$ at a given point $t_0 \in (0, \infty)$. This result describes explicitly the roles played by the parameters $α, γ$, and the location $t_0$. Our theorem differs from the earlier Strassen's {global law of the iterated logarithm} for $X$ proved by Ichiba, Pang and Taqqu (2022). |
| title | Strassen's local law of the iterated logarithm for the generalized fractional Brownian motion |
| topic | Probability 60G15, 60G17, 60G18, 60G22 |
| url | https://arxiv.org/abs/2411.15681 |