A new limit theorem for the critical Bellman–Harris branching process

Let $z(t)$ be the number of particles in a Bellman–Harris process at time $t$, $G(t)$ the distribution function of the lifetimes of the particles, $f(s)$ the generating function of the number of offspring of one particle, and $f'(1)=1$.
In the case when $ f(s)=s+(1-s)^{1+\alpha}L(1-s)$, where $\alpha\in(0,1)$ and $L(x)$ is slowly varying as $x\to+0$, and $n(1-G(n))\sim c(1-f_n(0))$, as $n\to\infty$, it is shown that
$$ \lim_{t\to\infty}\mathsf P\{z(t)\varphi(t)\le x\mid z(t)> 0\} $$
for a function $\varphi(t)$ equal either to 1 or to $\mathsf P\{z(t)>0\}$.
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