Non-exctinction probability for two branching processes in a joint random environment

Abstract

The paper introduces the model of a pair of branching processes $\{\boldsymbol{Z_n} = \left(Z_n^{(1)}, Z_n^{(2)}\right), \; n \in \mathbb{N}_0\}$ in a joint random environment. If the environment is fixed then the sequences $\{Z_n^{(1)}, \; n \in \mathbb{N}_0\}$ and $\{Z_n^{(2)},\; n \in \mathbb{N}_0\}$ are independent branching processes in a varying environment. This model is a particular case of a more general model of a multitype branching process in a random environment. We establish the asymptotic relation ${\bf{P}}\left(Z_n^{(1)} > 0, Z_n ^{(2)}>0 \right) \sim C n^{-a}$ as $n \to \infty$, where the parameter $a$ depends only on the correlation coefficient $\rho$ of an increment of a two-dimensional associated random walk.