  We refine some previous results concerning the Renewal Contact Processes. We
significantly widen the family of distributions for the interarrival times for
which the critical value can be shown to be strictly positive. The result now
holds for any spatial dimension $d \geq 1$ and requires only a moment condition
slightly stronger than finite first moment. We also prove a Complete
Convergence Theorem for heavy tailed interarrival times. Finally, for heavy
tailed distributions we examine when the contact process, conditioned on
survival, can be asymptotically predicted knowing the renewal processes. We
close with an example of an interarrival time distribution attracted to a
stable law of index 1 for which the critical value vanishes, a tail condition
uncovered by previous results.
