A note on the stability of the local time of a wiener process

AbstractLet L(a, t) be the local time of a Wiener process, and put A(t) = sup|a|⩽g(t)L(a, t)L(0, t)−1. It is shown that if g(t)=t12(log t)−1(log log t)−1 and limt→∞ A(t)=0 a.s. when p > 2 and lim sumt→∞ A(t)=0 a.s. when p = 1. A similar result is proved for random g(t) depending on the maximum of the Wiener process. These results settle a problem posed by Csörgő and Révész [7]