Geodesic
On a~class of branching processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 1, pp. 182-187

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We study $\lim\limits_{n\to\infty}\mathbf P\{z_n=0\}$ (the probability of “degeneration”) where 1) $z_n=\sum\limits_{k=1}^{[z_{n-1}/a]}\xi_k+z_{n-1}-a[z_{n-1}/a]$, $n\ge1$ 2) $a$ is a positive integer; 3) $\xi_n\ge0$ $(n\ge1)$ is a sequence of independent identically distributed integer-valued random variables. If $a=1$, the sequence $\{z_n,n\ge0\}$ is an usual Galton–Watson branching process. 
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     title = {On a~class of branching processes},
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I. I. Ezhov; A. A. Shahbazov. On a~class of branching processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 1, pp. 182-187. http://geodesic.mathdoc.fr/item/TVP_1975_20_1_a20/