Large Favourite Sites of Simple Random Walk and the Wiener Process Large Favourite Sites of Simple Random Walk and the Wiener Process

Let U (n) denote the most visited point by a simple symmetric random walk fS k g k0 in the rst n steps. It is known that U (n) and max 0kn S k satisfy the same law of the iterated logarithm, but have diierent upper functions (in the sense of P. L evy). 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. Summary. Let U (n) denote the most visited point by a simple symmetric random walk fS k g k0 in the rst n steps. It is known that U (n) and max 0kn S k satisfy the same law of the iterated logarithm, but have diierent upper functions (in the sense of P. L evy). 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.