  The main result of this paper is to show that if $\H$ is a normal subgroup of
a Kleinian group $G$ such that $G/\H$ contains a coset which is represented by
some loxodromic element, then the Hausdorff dimension of the transient limit
set of $\H$ coincides with the Hausdorff dimension of the limit set of $G$.
This observation extends previous results by Fern\'andez and Meli\'an for
Riemann surfaces.
