Stationary Distribution and Limit Conditional Law in the Subcritical Markov Branching Processes with/without Homogeneous Immigration

Abstract The study examines the relationship between the Lévy measure of the immigration flux of particles, the branching mechanism, and the limit conditional law (LCL) of Markov branching processes (MBP). The aim is to obtain a given stationary distribution for a subcritical branching process with immigration. To achieve this, the work focuses on the particular cases in which the LCL is defined by the Borel, logarithmic, and geometric distributions. The convergence of the extended Sibuya law to the logarithmic series is also discussed. In particular, when the Lévy measure of the immigration coincides with the LCL, the stationary distribution is a compound Poisson law with the same Lévy measure. The Neyman–Scott (NS) immigration process provides a counterexample to this case, because its Lévy measure given by the zero-truncated Poisson distribution is log-concave. The stationary distribution of the NS immigration process with linear death branching is characterized by the special function known as the modified exponential integral, namely Ein ⁡ ( z ) \operatorname{Ein}(z) . Some counterexamples are more interesting than examples.