On the asymptotic structure of non-critical Markov stochastic branching processes with continuous time

The work is devoted to the study of the transition probabilities of Markov branching random processes of continuous time with minimal moment conditions. Consider the non-critical case, i.e. the case when the average density of the conversion rate of particles is not zero. Let us find an asymptotic representation for the transition probabilities without additional moment conditions. To find the finite limiting invariant distribution, we restrict ourselves to the condition of finiteness of the moment of the type $\mathbb{E}[x \ln x]$ for the transformation density of particles. The statement on the asymptotic representation of the probabilistic generating function (Main Lemma) of the process under study and its differential analogue will underlie our conclusions. The theory of regularly varying functions in the sense of Karamat is essentially applied.