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A new limit theorem for the critical Bellman–Harris branching process
Sbornik. Mathematics, Tome 37 (1980) no. 3, pp. 411-423
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Let $z(t)$ be the number of particles in a Bellman–Harris process at time $t$, $G(t)$ the distribution function of the lifetimes of the particles, $f(s)$ the generating function of the number of offspring of one particle, and $f'(1)=1$. In the case when $ f(s)=s+(1-s)^{1+\alpha}L(1-s)$, where $\alpha\in(0,1)$ and $L(x)$ is slowly varying as $x\to+0$, and $n(1-G(n))\sim c(1-f_n(0))$, as $n\to\infty$, it is shown that $$ \lim_{t\to\infty}\mathsf P\{z(t)\varphi(t)\le x\mid z(t)> 0\} $$ for a function $\varphi(t)$ equal either to 1 or to $\mathsf P\{z(t)>0\}$. Bibliography: 11 titles. 
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     title = {A new limit theorem for the critical {Bellman{\textendash}Harris} branching process},
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V. A. Vatutin. A new limit theorem for the critical Bellman–Harris branching process. Sbornik. Mathematics, Tome 37 (1980) no. 3, pp. 411-423. http://geodesic.mathdoc.fr/item/SM_1980_37_3_a8/

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