We determine the tail asymptotics of the stationary distribution of a branching process with immigration in a random environment, when the immigration distribution dominates the offspring distribution. The assumptions are the same as in the Grincevi\v{c}ius--Grey theorem for the stochastic recurrence equation.
Moments, moderate and large deviations for a branching process in a random environment
Cell contamination and branching process in random environment with immigration
Branching Processes in Random Environments with Thresholds
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