Asymptotic properties of the stationary measure of a Markov branching process

Abstract

The asymptotic properties of the unique stationary measure of a Markov branching process will be given. In the critical case with finite variance, the result can be deduced from a result for discrete time processes of Kesten, Ney and Spitzer (1966) where the proof makes use of a stronger assumption than the finiteness of variance. For the continuous time case where the stationary measure has an explicit form, we can use the discrete renewal theorem which takes care of the infinite variance case as well.