On a class of limit theorems for a critical Bellman–Harris branching process

Let $z(t)$ be a critical Bellman–Harris branching process with lifetime distribution $G(t)$ and offspring generating function $f(s)=s+(1-s)^{1+\alpha}L(1-s)$, where $0<\alpha\le 1$ and $L(s)$ is slowly varying at 0. Let us denote by $f_k(s)$ the $k$-th iterate of $f(s)$. For the case when
$$ 0\le\liminf_{n\to\infty}\frac{n(1-G(n))}{1-f_n(0)}<\limsup_{n\to\infty}\frac{n(1-G(n))}{1-f_n(0)}<\infty $$
we prove some limit theorems for the process $z(t)$ which are analogous to those in [3].