The asymptotic behavior of the tail probabilities for the first hitting times of the Bessel process with arbitrary index is shown without using the explicit expressions for the distribution function obtained in the authors' previous works.
"Asymptotics of the probability distributions of the first hitting times of Bessel processes." Electron. Commun. Probab. 19 1 - 5, 2014. https://doi.org/10.1214/ECP.v19-3215