Sufficient conditions for the existence of a limiting distribution for a multitype branching process with immigration in a random environment, $Z(t)$, are given. In the case when the environment is an independent, identically distributed sequence, sufficient conditions are given which insure that the tail of the distribution of ν = inf > 0: Z(t) = 0\ decreases exponentially fast, and an application of this fact to random walk in a random environment is indicated.
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Eric S. Key (1987) studied this question.