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Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero

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Abstract

We consider the subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are change of measure and some conditional limit theorems for random walks.

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Acknowledgements

The authors are deeply grateful to an anonymous referee for careful reading of the original manuscript and for helpful suggestions allowing us to improve the presentation of the paper significantly.

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Correspondence to Vladimir Vatutin.

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Doudou Li and Mei Zhang were supported by the Natural Science Foundation of China under the Grant 11871103, and V. Vatutin was partially supported by the High-End Foreign Experts Recruitment Program (No. GDW20171100029).

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Li, D., Vatutin, V. & Zhang, M. Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero. J Theor Probab (2020). https://doi.org/10.1007/s10959-020-00991-5

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Keywords

  • Branching processes
  • Random environment
  • Immigration
  • Life period

Mathematics Subject Classification (2010)

  • 60J80
  • 60F99