Abstract:
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, immigration, conditioned random walk.
Citation:
V. A. Vatutin, C. Smadi, “Critical Branching Processes in a Random Environment with Immigration: The Size of the Only Surviving Family”, Branching Processes and Related Topics, Collected papers. On the occasion of the 75th birthday of Andrei Mikhailovich Zubkov and 70th birthday of Vladimir Alekseevich Vatutin, Trudy Mat. Inst. Steklova, 316, Steklov Math. Inst., Moscow, 2022, 355–375; Proc. Steklov Inst. Math., 316 (2022), 336–355
\Bibitem{VatSma22}
\by V.~A.~Vatutin, C.~Smadi
\paper Critical Branching Processes in a Random Environment with Immigration: The Size of the Only Surviving Family
\inbook Branching Processes and Related Topics
\bookinfo Collected papers. On the occasion of the 75th birthday of Andrei Mikhailovich Zubkov and 70th birthday of Vladimir Alekseevich Vatutin
\serial Trudy Mat. Inst. Steklova
\yr 2022
\vol 316
\pages 355--375
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4206}
\crossref{https://doi.org/10.4213/tm4206}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2022
\vol 316
\pages 336--355
\crossref{https://doi.org/10.1134/S0081543822010230}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85129013723}
Citation in format AMSBIB
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