A problem of Foldes and Puri on the Wiener process

Abstract:

Let $W$ be a real-valued Wiener process starting from 0, and $\tau (t)$ be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of $X(t)=\int _0^{\tau (t)} \mathbb {1}_{\{|W(u)| \le \alpha t\}}\,du$ as $t$ tends to infinity, i.e. they ask: how long does $W$ stay in a tube before “crossing very much” a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for $X$.