Markov Branching Process with Infinite Variance and Poisson Immigration with Decreasing Intensity

Abstract

We consider critical Markov branching processes with infinite variance of the offspring distribution which admit an immigration 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 zero. The probability for non-hitting zero is asymptotically investigated and limiting distributions are obtained, under suitable normalization.