On the number of productive individuals in the Bienayme–Galton–Watson process with immigration

Let Z(τ,t) is the number of individuals at time τ having more than θ(t−τ) descendants at time t,t>τ. Here θ(t) is some non-negative function. Limit distributions for Z(τ,t) when population evolves according to critical branching processes with time homogeneous immigration and distribution of the number of descendants has finite variance are obtained. An application to study of the number of “big” trees in a forest containing a random number of trees is also discussed.