A problem of Földes and Puri on the Wiener process

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 $X(t)=\int _0^{\tau (t)} {\text{\bf 1}\hskip -1.25pt\mathrm{l}}_{\{ | W(u)| \le \alpha t\} }du$ 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 $X$.