Moments of multitype critical Bellman–Harris processes in which tails of life-length distributions of particles have different orders

A multitype indecomposable, nonperiodic, critical Bellman–Harris branching process is considered. It is assumed that the types of the process may be splitted into two classes. A particle whose type belongs to the first class has a finite expected life-length, while the expected life-length of a particle whose type belongs to the second class is infinite.
Assuming that the tail of the life-length distribution of a particle with type from the second class is regularly varying at infinity with parameter depending on the type, we investigate the asymptotic behavior of the first and second moments for the number of particles of all types as well as the increments of the first moments. Our proofs are based on the asymptotic properties of some renewal matrices defined in terms of certain characteristics of the initial Bellman–Harris branching process.