Large Favourite Sites of Simple Random Walk and theWiener Process

Abstract

Let $U(n)$ denote the most visited point by a simple symmetric random walk $\{ S_k\}_{k\ge 0}$ in the first $n$ steps. It is known that $U(n)$ and $max_{0\le k\le n} S_k$ satisfy the same law of the iterated logarithm, but have different upper functions (in the sense of P. Lévy). The distance between them however turns out to be transient. In this paper, we establish the exact rate of escape of this distance. The corresponding problem for the Wiener process is also studied.