Let $R(n)$ be the largest integer for which the disc of radius $R(n)$ around the origin is covered by the first $n$ steps of a random walk. The main objective of the present paper is to obtain better estimates for the upper tail of the distribution of $R(n)$. For example, we show that there are constants $0 < \lambda_2 < \lambda_1 < \infty$ such that \begin{align*}\exp(-\lambda_1 z) &\leq \lim \inf_{n\rightarrow\infty} \mathbf{P}\big\{\frac{(\log R(n))^2}{\log n} > z\big\} \\ &\leq \lim \inf_{n\rightarrow\infty}\mathbf{P}\big\{\frac{(\log R(n))^2}{\log n} > z\big\} \leq \exp(-\lambda_2z).\end{align*}
Publié le : 1990-10-14
Classification:
Random walk,
covered disc,
local time,
limit distributions,
strong theorems,
60J15,
60F05
@article{1176990648,
author = {Revesz, P.},
title = {Estimates of the Largest Disc Covered by a Random Walk},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 1784-1789},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990648}
}
Revesz, P. Estimates of the Largest Disc Covered by a Random Walk. Ann. Probab., Tome 18 (1990) no. 4, pp. 1784-1789. http://gdmltest.u-ga.fr/item/1176990648/