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Under mild non-degeneracy assumptions on branching rates in each generation,we provide a criterion for almost-sure extinction of a multi-type branchingprocess with time-dependent branching rates. We also provide a criterion forthe total number of particles (conditioned on survival and divided by theexpectation of the resulting random variable) to approach an exponential randomvariable as time goes to infinity.

Multi-type branching processes with time dependent branching rates