Reduced branching processes with very heavy tails

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter $\alpha \in (0,1]$. We turn to the case of very heavy tailed reproduction distribution $\alpha =0$ assuming Zubkov’s regularity condition with parameter $\beta \in (0,\mathrm{\infty })$. Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.