Geodesic
Branching processes with final types of particles and random trees
Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 4, pp. 699-715

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This paper considers a Bellman–Harris branching process whose probability generating function $f(s)$ of the number of direct descendants of particles satisfies the relation $f(s) = s + (1 - s)^{1 + \alpha } L(1 - s)$, $0 \alpha \le 1$. Let $\tau $ be the moment of extinction of the process and let $\nu_\Delta $ be the total number of particles the number of direct descendants of each of which belongs to the set $\Delta ,\Delta \subset \{ 0,1, \ldots ,n, \ldots \} $. The paper gives conditions under which, for any $x \in ( - \infty , + \infty )$ and some scaling constants $b(N)$, a nondegenerate limit, $\lim _{N \to \infty } \mathbf{P}\{ \tau b(N) \le x\mid\nu_\Delta = N\} $, exists.
Keywords: Bellman–Harris branching process, a rooted random tree, the weight and height of a tree, limiting distributions
Mots-clés : final particles. 
@article{TVP_1994_39_4_a3,
     author = {V. A. Vatutin},
     title = {Branching processes with final types of particles and random trees},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {699--715},
     publisher = {mathdoc},
     volume = {39},
     number = {4},
     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1994_39_4_a3/}
}
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V. A. Vatutin. Branching processes with final types of particles and random trees. Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 4, pp. 699-715. http://geodesic.mathdoc.fr/item/TVP_1994_39_4_a3/