Long-time behavior of Markov branching processes controlled by Markov chains

It is well known that the evolution of a branching system is controlled by its branching mechanism. However, in realistic situations, the branching mechanism at different times may be different. In this paper, we mainly consider a special Markov branching process Z_n:ngeq 0, whose branching mechanism is controlled by a Markov chain ξ=ξn:ngeq 0on a branching mechanism space mathcalD The probability generating function, mathematical expectation, and variance of Z_n are presented. Moreover, it is proved that there exist a positive constant λ and a function v(·) such that fracv(ξn)Z_nλ^n