Reduced branching processes with very heavy tails

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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 α∈(0,1]. We turn to the case of very heavy tailed reproduction distribution α=0 assuming Zubkov's regularity condition with parameter β∈(0,∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.