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

The continuous-time Markov Branching Process with Immigration is discussed in the paper. A critical case wherein the second moment of offspring law and the first moment of immigration law are possibly infinite is considered. Assuming that the non-linear parts of the appropriate generating functions are regularly varying in the sense of Karamata, theorems on convergence of transition functions of the process to invariant measures are proved. The rate of convergence is determined provided that slowly varying factors are with remainder.