  We investigate subcritical Galton-Watson branching processes with immigration
in a random environment. Using Goldie's implicit renewal theory we show that
under general Cram\'er condition the stationary distribution has a power law
tail. We determine the tail process of the stationary Markov chain, prove point
process convergence, and convergence of the partial sums. The original
motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which
connects a random walk in a random environment model to a special Galton-Watson
process with immigration in a random environment. We obtain new results even in
this very special setting.
