Moments and distribution of the local times of a transient random walk on $\Z^d$
| dc.creator | Becker, Mathias | |
| dc.creator | Konig, Wolfgang | |
| dc.date | 2007-08-31 | |
| dc.date | 2008-05-07 | |
| dc.date.accessioned | 2026-07-07T09:37:10Z | |
| dc.date.available | 2026-07-07T09:37:10Z | |
| dc.description | Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $α\in[0,\infty)$ and let $L_n(α)$ be the spatial sum of the $α$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $α$, $L_n(α)$ is the number of the $α$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(α)$ as $n\to\infty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by Černý \cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/0708.4408 | |
| dc.identifier | http://arxiv.org/abs/0708.4408 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160362 | |
| dc.subject | Probability | |
| dc.subject | 60G50, 60J55, 60F15 | |
| dc.title | Moments and distribution of the local times of a transient random walk on $\Z^d$ | |
| dc.type | text |