  We study random walks on relatively hyperbolic groups whose law is
convergent, in the sense that the derivative of its Green function is finite at
the spectral radius.When parabolic subgroups are virtually abelian, we prove
that for such a random walk satisfies a local limit theorem of the form $p_n(e,
e)\sim CR^{-n}n^{-d/2}$, where $p_n(e, e)$ is the probability of returning to
the origin at time $n$, $R$ is the inverse of the spectral radius of the random
walk and $d$ is the minimal rank of a parabolic subgroup along which the random
walk is spectrally degenerate.This concludes the classification all possible
behaviour for $p_n(e, e)$ on such groups.
