  We consider a branching random walk with immigration in a random environment,
where the environment is a stationary and ergodic sequence indexed by time. We
focus on the asymptotic properties of the sequence of measures $(Z_n)$ that
count the number of particles of generation $n$ located in a Borel set of real
line. In the present work, a series of limit theorems related to the above
counting measures are established, including a central limit theorem, a
moderate deviation principle and a large deviation result as well as a
convergence theorem of the free energy.
