  We generalize the notion of cusp excursion of geodesic rays by introducing
for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic
$N$-manifold of finite volume. We show that on one hand, this excursion is at
most linear for geodesics that are generic with respect to the hitting measure
of a random walk. On the other hand, for $k = N-1$, the $k^{th}$ excursion is
superlinear for geodesics that are generic with respect to the Lebesgue
measure. We use this to show that the hitting measure and the Lebesgue measure
on the boundary of hyperbolic space $\mathbb{H}^N$ for any $N \geq 2$ are
mutually singular.
