Geodesic
Theorem on Sums of Independent Random Positive Variables and its Applications to Branching Processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 4, pp. 710-718

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Let $\xi_1,\dots,\xi_n,\dots$ be independent random positive variables and let ${\mathbf P}\{\xi_k$, $k=1,\dots,n,\dots$ Let us denote $$ {\mathbf P}\{\xi_1+\dots+\xi_n\}=G_n(t). $$ 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. 
@article{TVP_1964_9_4_a12,
     author = {V. P. \v{C}istyakov},
     title = {Theorem on {Sums} of {Independent} {Random} {Positive} {Variables} and its {Applications} to {Branching} {Processes}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {710--718},
     publisher = {mathdoc},
     volume = {9},
     number = {4},
     year = {1964},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a12/}
}
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V. P. Čistyakov. Theorem on Sums of Independent Random Positive Variables and its Applications to Branching Processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 4, pp. 710-718. http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a12/