Limsup Random Fractals
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | Limsup Random Fractals |
| 2. | Creator | Author's name, affiliation, country | Davar Khoshnevisan; University of Utah |
| 2. | Creator | Author's name, affiliation, country | Yuval Peres; University of California, Berkeley |
| 2. | Creator | Author's name, affiliation, country | Yimin Xiao; University of Utah |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | Limsup random fractal, packing dimension, Hausdorffdimension, Brownian motion, fast point. |
| 3. | Subject | Subject classification | 60G17, 69J65, 28A80. |
| 4. | Description | Abstract | Orey and Taylor (1974) introduced sets of ``fast points'' where Brownian increments are exceptionally large, ${\rm F}(\lambda):=\{ t\in[0,1]: \limsup_{h\to 0}{ | X(t+h)-X(t)| / \sqrt{ 2h|\log h|}} \ge \lambda\}$. They proved that for $\lambda \in (0,1]$, the Hausdorff dimension of ${\rm F}(\lambda)$ is $1-\lambda^2$ a.s. We prove that for any analytic set $E \subset [0,1]$, the supremum of the $\lambda$ such that $E$ intersects ${\rm F}(\lambda)$ a.s. equals $\sqrt{\text{dim}_p E }$, where $\text{dim}_p E$ is the packing dimension of $E$. We derive this from a general result that applies to many other random fractals defined by limsup operations. This result also yields extensions of certain ``fractal functional limit laws'' due to Deheuvels and Mason (1994). In particular, we prove that for any absolutely continuous function $f$ such that $f(0)=0$ and the energy $\int_0^1 |f'|^2 \, dt $ is lower than the packing dimension of $E$, there a.s. exists some $t \in E$ so that $f$ can be uniformly approximated in $[0,1]$ by normalized Brownian increments $s \mapsto [X(t+sh)-X(t)] / \sqrt{ 2h|\log h|}$; such uniform approximation is a.s. impossible if the energy of $f$ is higher than the packing dimension of $E$. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | |
| 7. | Date | (YYYY-MM-DD) | 2000-02-09 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ejp.ejpecp.org/article/view/60 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v5-60 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 5 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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