We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of valleys of height $log t$. In t
he quenched setting, we also sharply estimate the distribution of the walk at time $t$.
We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian moti
on, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.
