The escape rate of favorite sites of simple random walk and Brownian motion

Consider a simple symmetric random walk on the integer lattice Z. For each n, let V(n) denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat Surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417-436] says that V is almost surely transient, thus disproving a previous conjecture of Erdos and Revesz [Mathematical Structures-Computational Mathematics-Mathematical Modeling 2 (1984) 152-157]. More precisely, Bass and Griffin proved that almost surely, lim inf(n-->infinity)(\V(n)\)/(1/2)(-gamma)(n)((logn)) equals 0 if gamma < 1, and infinity if gamma > 11 (eleven). The present paper studies the rate of escape of V(n). We show that almost surely, the "lim inf"' expression in question is 0 if y < 1, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.