Critical Markov branching process with infinite variance allowing Poisson immigration with increasing intensity

The article studies a single-type critical Markov branching process with infinite variance of the offspring distribution. The process admits also an immigration component at the time points of a non-homogeneous Poisson process. If additionally the mean number of immigrants is infinite, then proper limit distributions are obtained, under suitable normalization of the sample paths, depending on the rate of change of the intensity of the Poisson process.