On estimation of the convergence rate to invariant measures in markov branching processes with possibly infinite variance and allowing immigration

The paper discusses the continuous-time Markov Branching Process allowing Immigration. We are considering a critical case for which the second moment of offspring law and the first moment of immigration law are possibly infinite. Assuming that the nonlinear parts of the appropriate generating functions are regularly varying in the sense of Karamata, we prove theorems on convergence of transition functions of the process to invariant measures. We deduce the speed rate of these convergence providing that slowly varying factors are with the remainder.