For a system of infinitely many independent symmetric random walks on $\mathbb{Z}$ let $K_n(x)$ be the number of visits to $x \in \mathbb{Z}$ from time 0 to $n - 1$. The probabilities of some rare events involving $(K_n(0), K_n(1))$ are estimated as $n \rightarrow \infty$ and the corresponding large deviation rate functions are derived for both deterministic and invariant initial distributions. The dependence on the initial distributions is discussed. A simple method is used for guessing at the rate functions. This method is effective for independent random walks on the line and is worth exploring in more general settings.
Publié le : 1995-07-14
Classification:
Large deviations,
occupation time,
random walk,
infinite particle system,
60B12,
60F05,
60F10,
60J15
@article{1176988186,
author = {Lee, Tzong-Yow},
title = {Large Deviations for Independent Random Walks on the Line},
journal = {Ann. Probab.},
volume = {23},
number = {3},
year = {1995},
pages = { 1315-1331},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988186}
}
Lee, Tzong-Yow. Large Deviations for Independent Random Walks on the Line. Ann. Probab., Tome 23 (1995) no. 3, pp. 1315-1331. http://gdmltest.u-ga.fr/item/1176988186/