Geodesic
On a~class of limit theorems for a~critical Bellman--Harris branching process
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 818-824

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Let $z(t)$ be a critical Bellman–Harris branching process with lifetime distribution $G(t)$ and offspring generating function $f(s)=s+(1-s)^{1+\alpha}L(1-s)$, where $0\alpha\le 1$ and $L(s)$ is slowly varying at 0. Let us denote by $f_k(s)$ the $k$-th iterate of $f(s)$. For the case when $$ 0\le\liminf_{n\to\infty}\frac{n(1-G(n))}{1-f_n(0)}\limsup_{n\to\infty}\frac{n(1-G(n))}{1-f_n(0)}\infty $$ we prove some limit theorems for the process $z(t)$ which are analogous to those in [3]. 
@article{TVP_1981_26_4_a12,
     author = {V. A. Vatutin},
     title = {On a~class of limit theorems for a~critical {Bellman--Harris} branching process},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {818--824},
     publisher = {mathdoc},
     volume = {26},
     number = {4},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_4_a12/}
}
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V. A. Vatutin. On a~class of limit theorems for a~critical Bellman--Harris branching process. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 818-824. http://geodesic.mathdoc.fr/item/TVP_1981_26_4_a12/