Number Variance for Hierarchical Random Walks and Related Fluctuations
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | Number Variance for Hierarchical Random Walks and Related Fluctuations |
| 2. | Creator | Author's name, affiliation, country | Tomasz Bojdecki; University of Warsaw; Poland |
| 2. | Creator | Author's name, affiliation, country | Luis G. Gorostiza; Centro de Investigacion y de Estudios Avanzados Mexico; Mexico |
| 2. | Creator | Author's name, affiliation, country | Anna Talarczyk; University of Warsaw; Poland |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | hierarchical random walk; hierarchical group; ultrametric; number variance; fluctuation; limit theorem |
| 3. | Subject | Subject classification | 60G50; 60F05 |
| 4. | Description | Abstract | We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the c-random walks. Such walks are used, e.g., in mathematical physics and population biology. The number variance problem consists in investigating if the variance of the number of “particles” $N_n(L)$ lying in the ball of radius $L$ at a given step $n$ remains bounded, or even better, converges to a finite limit, as $L\to\infty$. We give a necessary and sufficient condition and discuss its relationship to transience/recurrence property of the walk. Next we consider normalized fluctuations of $N_n(L)$ around the mean as $n\to\infty$ and $L$ is increased in an appropriate way. We prove convergence of finite dimensional distributions to a Gaussian process whose properties are discussed. As the $c$-random walks mimic symmetric stable processes on $\mathbb{R}$, we compare our results with those obtained by Hambly and Jones (2007, 2009), who studied the number variance problem for an infinite system of independent symmetric stable processes on $\mathbb{R}$. Since the hierarchical group is an ultrametric space, corresponding results for symmetric stable processes and hierarchical random walks may be analogous or quite different, as has been observed in other contexts. An example of a difference in the present context is that for the stable processes a fluctuation limit process is a Gaussian process which is not Markovian and has long range dependent stationary increments, but the counterpart for hierarchical random walks is Markovian, and in a special case it has independent increments. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | CONACyT grant 98998 (Mexico) and MNiSzW grant N N201 397537 (Poland) |
| 7. | Date | (YYYY-MM-DD) | 2011-10-31 |
| 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/937 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v16-937 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 16 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
| 15. | Rights | Copyright and permissions | The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available. Summary of the Creative Commons Attribution License You are free
|