Abstract
We consider a critical branching process \(\{Y_n,\,n\geq0\}\) 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 \(n\)).
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Acknowledgments
The authors thank the anonymous referee, whose comments helped to eliminate a number of inaccuracies contained in the original version of the paper.
Funding
This work is supported by the Russian Science Foundation under grant 19-11-00111.
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 316, pp. 355–375 https://doi.org/10.4213/tm4206.
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Vatutin, V.A., Smadi, C. Critical Branching Processes in a Random Environment with Immigration: The Size of the Only Surviving Family. Proc. Steklov Inst. Math. 316, 336–355 (2022). https://doi.org/10.1134/S0081543822010230
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DOI: https://doi.org/10.1134/S0081543822010230