Properties of multitype subcritical branching processes in random environment

V. A. Vatutin ; E. E. D'yakonova

Diskretnaya Matematika, Tome 32 (2020) no. 3, pp. 3-23.

Résumé

We study properties of a $p$-type subcritical branching process in random environment initiated at moment zero by a vector $\mathbf{z}=\left( z_{1},..,z_{p}\right) $ of particles of different types. For $p=1$ the class of processes we consider corresponds to the so-called strongly subcritical case. It is shown that the survival probability of this process up to moment $n$ behaves as $C(\mathbf{z})\lambda ^{n}$ for large $n,$ where the parameters $\lambda\in (0,1) $ and $C(z)\in (0,\infty)$ are explicitly described in terms of the characteristics of the process. We also demonstrate that the distribution of the number of particles of different types at moment $n\rightarrow \infty$ (given its survival up to this moment) does not asymptotically depend on the number and types of particles initiated the process.

Détail

Keywords: random environment, multitype branching processes, limit theorems.

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