  It is known that there exists the stationary sequence of branching process
with immigration $\{X_{n}\}_{n\in\mathbb{Z}}$ under some conditions (Foster and
Williamson (1971)), when the offspring is critical or subcritical. A precise
large deviation probability for the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ is
specified, the significant difference is revealed for the critical and
subcritical cases.
