Asymptotics Of The Invariant Measure Of The Generalized Markov Branching Processes

Markov branching processes play an important role in the theory and applications of stochastic processes. The regularity, uniqueness, recurrence and ergodicity are classical topics for the study of the branching processes. While in this thesis, we study the asymptotics of the invariant measures and distributions, a relatively new topic without much literature for Markov branching processes.In this thesis, we consider a generalized Markov branching process with resurrection, and investigate the asymptotics of the invariant measure in terms of its generating functions and the complex analysis methods (e.g. the Pringsheim and Tauberian theorems). By setting up the connection between the invariant measure of a Markov branching process and the invariant measure of a continuous-time random walk, we convert the study of asymptotics of the invariant measure for the Markov branching process to the asymptotic analysis of the invariant measure for the Q-matrix of the random walk. The subsequent analysis is divided into two cases according to whether the invariant measure of the Q-matrix is summable or not. We obtain the tail decay rate of the invariant distribution in the summable case and the divergence rate of the invariant measure in the non-summable case. The results are further applied to investigate the asymptotics of a truncated model.