Critical Bellman–Harris branching processes with long-living particles

A critical indecomposable two-type Bellman–Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter $\beta\in(0,1]$. It is shown that, contrary to the critical indecomposable Bellman–Harris branching processes with finite variances of the life-lengths of particles of both types, the probability of observing first-type particles at a distant moment $t$ is infinitesimally less than the survival probability of the whole process. In addition, a Yaglom-type limit theorem is proved for the distribution of the number of the first-type particles at moment $t$ given that the population contains particles of the first type at this moment.