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We determine the tail asymptotics of the stationary distribution of abranching process with immigration in a random environment, when theimmigration distribution dominates the offspring distribution. The assumptionsare the same as in the Grincevi\v{c}ius--Grey theorem for the stochasticrecurrence equation.

Branching Processes with Immigration in a Random Environment -- the Grincevi\v{c}ius--Grey setup