Poisson Random Measures and Noncritical Multitype Markov Branching Processes

We investigate noncritical multi-type Markov branching processes with immigration generated by Poisson measures. Limiting distributions are obtained when the rates of the Poisson measures are asymptotically equivalent to exponential or regularly varying functions. In particular, results analogous to a strong LLN are presented, and limiting normal distributions are obtained when the rates increase. When the rates decrease, then conditional limiting distributions are established. A stationary limiting distribution is obtained when the mean Poisson measure grows linearly. The asymptotic behaviour of the first and second moments of the processes is also investigated. NFSR of the Ministry of Education and Science of Bulgaria, grant No. KP–06-H22/3.