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

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    
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    
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    

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