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Scaling limits for continuous-time branching processes with discrete statespace are provided as the initial state tends to infinity. Depending on thefiniteness or non-finiteness of the mean and/or the variance of the offspringdistribution, the limits are in general time-inhomogeneous Gaussian processes,time-inhomogeneous generalized Ornstein-Uhlenbeck type processes orcontinuous-state branching processes. We also provide transfer results showinghow specific asymptotic relations for the probability generating function ofthe offspring distribution carry over to those of the one-dimensionaldistributions of the branching process.

Asymptotics of continuous-time discrete state space branching processes for large initial state