An almost sure invariance principle for the range of planar random walks
| dc.creator | Bass, Richard F. | |
| dc.creator | Rosen, Jay | |
| dc.date | 2004-04-04 | |
| dc.date.accessioned | 2026-07-07T05:07:05Z | |
| dc.date.available | 2026-07-07T05:07:05Z | |
| dc.description | For a symmetric random walk in $Z^2$ with $2+δ$ moments, we represent $|\mathcal{R}(n)|$, the cardinality of the range, in terms of an expansion involving the renormalized intersection local times of a Brownian motion. We show that for each $k\geq 1$ \[ (\log n)^k [ \frac{1}{n} |\mathcal{R}(n)| +\sum_{j=1}^k (-1)^j (\textstyle{\frac1{2π}}\log n +c_X)^{-j} γ_{j,n}]\to 0, \qquad a.s. \] where $W_t$ is a Brownian motion, $W^{(n)}_t=W_{nt}/\sqrt n$, $γ_{j,n}$ is the renormalized intersection local time at time 1 for $W^{(n)}$, and $c_X$ is a constant depending on the distribution of the random walk. | |
| dc.identifier | https://arxiv.org/abs/math/0404070 | |
| dc.identifier | http://arxiv.org/abs/math/0404070 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70722 | |
| dc.subject | Probability | |
| dc.subject | 60G50; 60J65 | |
| dc.title | An almost sure invariance principle for the range of planar random walks | |
| dc.type | text |