Non-extinction of a pair of branching processes in a common random environment

We introduce a 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 common random environment (PBPRE) in our report. We assume that for the fixed environment the sequences $\{Z_{n, 1}, \; n \in \mathbb{N}_0\}$ and $\{Z_{n, 2},\; n \in \mathbb{N}_0\}$ are independent branching process in a varying environment.
PBPRE is a particular case of multitype branching process in a random environment (MBPRE). However, in MBPRE the particles of one type produce the particles of other types. In our case it's forbidden. This simplification orders to study the process under mild assumptions.
We consider a critical PBPRE $\{\boldsymbol{Z_n}\}$, meaning that both the process $\{Z_{n,1}\}$ and the process $\{Z_{n,2}\}$ are assumed to be critical. By non-extinction of the process $\{\boldsymbol{Z_n}\}$ we mean the non-extinction of both particle types. We will show that the following asymptotic relation holds:
$$\mathbf{P}\left(Z_{n, 1} > 0, Z_{n,2}>0 \right) \sim C n^{-a}, \quad n \to \infty,$$
where the parameter $a$ depends only on the correlation coefficient $\rho$ of an increment of a two-dimensional associated random walk. It is assumed that the correlation coefficient $\rho$ belongs to the interval $(0, 1)$. Also some restrictions of an associated random walk increment are imposed.
As in the case of a branching process in a random environment, the non-extinction probability of PBPRE to the moment $n$ differs only by a multiplicative constant from the probability of the event that the associated random walk stays “positive” as $n \to \infty$.
Note that D. Denisov and V. Watchel in [1] shed the light on aspects related to the “positivity” of multidimensional random walks.

• Denisov D., Wachtel V. Random walks in cones revisited //Annales de l'Institut Henri Poincare (B) Probabilites et statistiques. – Institut Henri Poincaré, 2024. – Т. 60. – №. 1. – С. 126-166.