Limit Theorems for Subcritical Age-dependent Branching Processes with Two Types of Immigration

For the classical subcritical age-dependent branching process the effect of the following two-type immigration pattern is studied. At a sequence of renewal epochs a random number of immigrants enters the population. Each subpopulation stemming from one of these immigrants or one of the ancestors is revived by new immigrants and their offspring whenever it dies out, possibly after an additional delay period. All individuals have the same lifetime distribution and produce offspring according to the same reproduction law. This is the Bellman-Harris process with immigration at zero and immigration of renewal type (BHPIOR). We prove a strong law of large numbers and a central limit theorem for such processes. Similar conclusions are obtained for their discrete-time counterparts (lifetime per individual equals one), called GaltonWatson processes with immigration at zero and immigration of renewal type (GWPIOR). Our approach is based on the theory of regenerative processes, renewal theory and occupation measures and is quite different from those in earlier related work using analytic tools.