Total Progency in a Critical Age-Dependent Branching Process with Immigration

Let Zt total number of cells born by t to critical age-dependent branching processes initiated by immigration at renewal epochs, with a random number of new cells introduced at each epoch, and where the mean cell lifetime is unequal to that of the mean time between immigration epochs. By extending a result for the discrete-time case of Pakes, and using approximation techniques and a rate of convergence result for a generalized law of large numbers, it is shown that, using second moments that the limit of E as t approaches infinity for exp- theta t square zt exists, and is explicitly given.