The total number of particles in a reduced Bellman–Harris branching process

Let $z(t)$ be the number of particles at time $t$ in a Bellman-Harris branching process with generating function $f(s)$ of the number of direct descendants and distribution $G(t)$ of particle lifelength satisfying the conditions
$$ f'(1) = 1,\qquad f(s) = s + (1 - s)^{1 + \alpha } L(1 - s), $$
where $\alpha \in ( {0,1} ]$, the function $L(x)$ varies slowly as $x \to 0 + $, and
$$ \lim_{n \to \infty }\frac{{n( {1 - G(n))}}} {{1 - f_n( 0 )}} = 0, $$
where $f_n ( s )$ is the $n$th iteration of $f(s)$. Denote by $\{ z(\tau ,t), 0 \le \tau \le t\}$ the corresponding reduced Bellman-Harris branching process, where $z(\tau ,t)$ is the number of particles in the initial process at time $\tau $ whose descendants or they themselves are alive at time $t$. Let $\nu (t)$ be the number of dead particles of the reduced process to time $t$. The paper finds the limiting distribution of $\nu(t)$ under the conditions $z(t) > 0$ and $t \to \infty $.