Geodesic
Subcritical branching processes in random environment with
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 671-692

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We consider a subcritical branching process in an independent and identically distributed (i.i.d.) random environment, where one immigrant arrives at each generation. We consider the event $\mathcal{A}_{i}(n)$ in which all individuals alive at time $n$ are descendants of the immigrant, who joined the population at time $i$, and investigate the asymptotic probability of this extreme event for $n\to \infty$ when $i$ is fixed, the difference $n-i$ is fixed, or $\min (i,n-i)\to \infty$. To deduce the desired asymptotics we establish some limit theorems for random walks conditioned to be nonnegative or negative on $[0,n]$.
Keywords: branching process, random environment, conditioned random walk.
Mots-clés : immigration 
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     author = {V. A. Vatutin and E. E. D'yakonova},
     title = {Subcritical branching processes in random environment with},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {671--692},
     publisher = {mathdoc},
     volume = {65},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a1/}
}
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V. A. Vatutin; E. E. D'yakonova. Subcritical branching processes in random environment with. Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 671-692. http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a1/