Estimates of the Largest Disc Covered by a Random Walk

Abstract

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*}