Theorem on Sums of Independent Random Positive Variables and its Applications to Branching Processes

Let $\xi_1,\dots,\xi_n,\dots$ be independent random positive variables and let ${\mathbf P}\{\xi_k<t\}=G(t)$, $k=1,\dots,n,\dots$ Let us denote
$$ {\mathbf P}\{\xi_1+\dots+\xi_n<t\}=G_n(t). $$

Theorem.
$$ \lim_{t\to\infty}\frac{1-G_n(t)}{1-G(t)}=n,\qquad n=1,2,3,\dots, $$
and only if
$$ \lim_{t\to\infty}\frac{1-G_2(t)}{1-G(t)}=2. $$
This theorem is useful in some investigations of age-dependent branching processes.