On a class of critical Markov branching processes with non-homogeneous Poisson immigration

The article studies a class of critical Markov branching processes with infinite variance of the offspring distribution. The processes admit also an immigration component at the jump-points of a non-homogeneous Poisson process, assuming that the mean number of immigrants is infinite and the intensity of the Poisson process converges to a constant. The asymptotic behavior of the probability for non-visiting zero is obtained. Proper limit distributions are proved, under suitable normalization of the sample paths, depending on the offspring distribution and the distribution of the immigrants.