Large favourite sites of simple random walk and the Wiener process. (English) Zbl 0908.60070

Csáki, Endre; Shi, Zhan

Electron. J. Probab. 3, Paper No. 14, 31 p. (1998).

Summary: Let \(U(n)\) denote the most visited point by a simple symmetric random walk \(\{ S_k\}_{k\geq 0}\) in the first \(n\) steps. It is known that \(U(n)\) and \(\max_{0\leq k\leq n} S_k\) 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. We establish the exact rate of escape of this distance. The corresponding problem for the Wiener process is also studied.

MSC:

60G50	Sums of independent random variables; random walks
60J65	Brownian motion
60J55	Local time and additive functionals

Keywords:

local time