  The main results in this paper concern large and moderate deviations for the
radial component of a $n$-dimensional hyperbolic Brownian motion (for $n\geq
2$) on the Poincar\'{e} half-space. We also investigate the asymptotic behavior
of the hitting probability $P_\eta(T_{\eta_1}^{(n)}<\infty)$ of a ball of
radius $\eta_1$, as the distance $\eta$ of the starting point of the hyperbolic
Brownian motion goes to infinity.
