Large deviations of branching processes with immigration in random environment

We consider branching process $Z_n$ in random environment such that the associated random walk $S_n$ has increments $\xi_i$ with mean $\mu$ and satisfy the Cramér condition $\mathbf{E}e^{h\xi_i}<\infty$, $0<h<h^+$. Let $\chi_i$ be the number of particles immigrating into the $i^{\rm th}$ generation of the process, $\mathbf{E}\chi_i^h<\infty$, $0<h<h^+$. We suppose that the number of offsprings of one particle conditioned on the environment has the geometric distribution. It is shown that the supplement of immigration to critical or supercritical processes results only in the change of multiplicative constant in the asymptotics of large deviation probabilities $\mathbf P\left\{Z_n\ge \exp(\theta n)\right\}$, $\theta>\mu$. In the case of subcritical processes analogous result is obtained for $\theta>\gamma$, where $\gamma>0$ is some constant. For all constants explicit formulas are given.