Sevastyanov branching processes with non-homogeneous Poisson immigration

Sevastyanov (Theory Probab Appl 2:339–348, 1957) introduced a class of Markov branching processes in which immigration of individuals in the population is allowed at random time points described by a time-homogeneous Poisson process. In the present paper, we study a model generalized this process along two directions: Sevastyanov’s (Theory Probab Appl 9:577–594, 1964) age-dependent branching process and time-nonhomogeneous Poisson immigration. The resulting process can be used to describe the dynamics of cell populations arising from differentiating stem cells. Limit theorems are proved in the supercritical case for various classes of immigration rates. Some of the limiting results offer generalizations of the classical result obtained in Sevastyanov (Theory Probab Appl 2:339–348, 1957). We also derive novel LLN and a CLT that arise from the fact that the process is time-inhomogeneous.

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