Geodesic
Critical Branching Processes in a Random Environment with Immigration: The Size of the Only Surviving Family
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 355-375

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We consider a critical branching process $\{Y_n,\,n\geq 0\}$ in an i.i.d. random environment in which one immigrant arrives at each generation. Let $\mathcal A_i(n)$ be the event that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$. We study the conditional distribution of $Y_n$ given $\mathcal A_i(n)$ when $n$ is large and $i$ follows different asymptotics which may be related to $n$ ($i$ fixed, close to $n$, or going to infinity but far from $n$).
Keywords: branching process, random environment, conditioned random walk.
Mots-clés : immigration 
@article{TM_2022_316_a22,
     author = {V. A. Vatutin and C. Smadi},
     title = {Critical {Branching} {Processes} in a {Random} {Environment} with {Immigration:} {The} {Size} of the {Only} {Surviving} {Family}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {355--375},
     publisher = {mathdoc},
     volume = {316},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_316_a22/}
}
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V. A. Vatutin; C. Smadi. Critical Branching Processes in a Random Environment with Immigration: The Size of the Only Surviving Family. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 355-375. http://geodesic.mathdoc.fr/item/TM_2022_316_a22/