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 motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.
"Renewal series and square-root boundaries for Bessel processes." Electron. Commun. Probab. 13 649 - 652, 2008. https://doi.org/10.1214/ECP.v13-1436