О номерах поколений в ветвящемся процессе с произвольным множеством типов частиц

The asymptotic behaviour of the ratio $\xi_t(S)$ of the number of particles, whose types are in a given set $S$ and ages do not exceed $at+zb\sqrt t$, where $t$ denotes the time and $a$ and $b$ are some parameters, to the total number of particles with the types in $S$ is studied. It is shown that, for supercritical processes,
$$ \xi_t(S)\underset{t\to\infty}\longrightarrow\Phi(z), $$
where $\Phi$ is the distribution function of $N(0,1)$.