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Scientific session of the Steklov Mathematical Institute of RAS dedicated to the results of 2020
November 25, 2020 15:45–16:00, Moscow, online
 


Branching process in random environment with immigration: life periods and survival of a single family

V. A. Vatutin, E. E. D'yakonova
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Abstract: We consider branching processes with immigration evolving in an i.i.d. random environment.
Assuming that immigration is not allowed when there are no individuals in the population we investigate in [1, 2] the tail distribution of the so-called life period of the subcritical and critical processes, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.
We also investigate in [3] a branching process with immigration 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 descendants of the immigrant which joined the population at time $i.$ Assuming that the process is subcritical we investigate, as $n\rightarrow \infty $ the asymptotic behavior of the probability of the event $\mathcal{A}_{i}(n)$ when $i$ is either fixed, or the difference $n-i$ is fixed, or $\min(i,n-i)\rightarrow \infty .$

References
  1. Doudou Li, Vladimir Vatutin, Mei Zhang, “Subcritical branching processes in random environment with immigration stopped at zero”, J. Theor. Probability, 2020, 1–23  mathnet  crossref
  2. Elena Dyakonova, Doudou Li, Vladimir Vatutin, Mei Zhang, “Branching processes in random environment with immigration stopped at zero”, J. Appl. Probab., 57:1 (2020), 237–249  mathnet  crossref
  3. V. A. Vatutin, E. E. Dyakonova, “Dokriticheskie vetvyaschiesya protsessy v sluchainoi srede s immigratsiei: vyzhivanie odnogo semeistva”, Teoriya veroyatn. i ee primen., 65:4 (2020), 671–692  mathnet  crossref


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