  We refine 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 dimension $d \ge 1$ and requires only a moment condition slightly
stronger than finite first moment. For heavy-tailed interarrival times, we
prove a Complete Convergence Theorem and examine when the contact process,
conditioned on survival, can be asymptotically predicted knowing the renewal
processes. We close with an example of distribution attracted to a stable law
of index 1 for which the critical value vanishes.
