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Seminar by Department of Discrete Mathematic, Steklov Mathematical Institute of RAS
February 25, 2025 16:00, Moscow, Steklov Mathematical Institute, Room 313 (8 Gubkina) + online


Some results for branching processes with weakly dependent immigration

S. O. Sharipov

V. I. Romanovskiy Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent



Abstract: Let $\left\{{{\xi_{k,j}},k,j \geq 1} \right\}$ and $\left\{{{\varepsilon_k},k\geq 1} \right\}$ be two sequence of non-negative integer-valued random variables such that the two families $\left\{ {{\xi_{k,j}}} \right\}$ and $\left\{ {{\varepsilon_k}} \right\}$ are independent, $\left\{ {{\xi_{k,j}}} \right\}$ are independent and identically distributed. We consider a sequence of branching processes with immigration $\left\{X_{k},k \geq 0\right\}$, defined by recursion:
$$ X_{0}= 0,\ \ X_{k}=\sum\limits_{j=1}^{X_{k-1}} {{\xi_{k,j}}+{\varepsilon_k}},\ \ k \geq 1. $$
We consider the critical case, i.e. when $m:=\mathbf{E}\xi_{1,1}=1$.
We primarily focus on the case where the immigration process follows non-identically distributed random variables. In the context of branching processes, this means that the immigration rate may depend on the time of immigration. I. Rahimov systematically investigated the asymptotic behaviour of $X_{k}$ in the case of increasing immigration (${\mathbb{E}}\varepsilon_{k}\to +\infty$ as $k \to \infty$).
Motivated by Rahimov's results on functional limit theorems for $X_{k}$ when the immigration sequence is independent, it is natural to ask about the generalization of these results to weakly dependent immigration sequences. In the talk, we also consider nearly critical branching processes with immigration, as well as partially observed branching processes.


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