Critical Age Dependent Branching Processes

Critical Age-dependent Branching Processes

Limit Probabilities for Critical Age-dependent Branching Processes with Immigration

Let Z sub O(t), N sub O (t) denote, respectively, the number of cells alive at t and the total progeny born by t in a process with a random number of new cells introduced at renewal epochs, each new cell initiating a critical age-dependent branching process. As t approaches infinity, the forms of P(Z sub O (t) = k) and P(N sub O (t) = k) are obtained for k = 1,2, ... and k = 0,1,2, ..., respectively. A multi-dimensional version and extension are indicated.

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

Let Z(t) = 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 z(t)) exists, and is explicitly given.