Seminars: V. I. Afanasyev, Branching processes in random environment initiated by a large number of initial particles

SEMINARS


Seminar by Department of Discrete Mathematic, Steklov Mathematical Institute of RAS September 24, 2024 16:00, Moscow, Steklov Mathematical Institute, Room 313 (8 Gubkina) + online

Branching processes in random environment initiated by a large number of initial particles V. I. Afanasyev Department of Discrete Mathematics, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Abstract: Let $\left\{ Z_{i},i=0,1,2,...\right\} $ be a critical branching process in random environment and the variance $\sigma ^{2}$ of a step of the associated random walk is finite and positive. Consider a sequence of branching processes $\mathbf{Z}^{(n,x)}=\left\{ Z_{i}^{(n,x)},i=0,1,...\right\} $, where $Z_{i}^{(n,x)}=\left\{ Z_{i}\|Z_{0}=m_{n}(x)\right\} $ and $\log m_{n}(x)\sim \sigma x\sqrt{n}$ as $ n\rightarrow \infty $ for some $x>0$. Three limit theorems are proved: on the extinction time of the process $ \mathbf{Z}^{(n,x)}$, on the properties of a normalized process constructed by $\mathbf{Z}^{(n,x)}$, and on the normalized logarithm of the process.