It is known that there exists the stationary sequence of branching process with immigration ₙ\ₙ under some conditions (Foster and Williamson (1971)), when the offspring is critical or subcritical. A precise large deviation probability for the partial sum Sₙ=X₁+⋯+Xₙ is specified, the significant difference is revealed for the critical and subcritical cases.
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Guo et al. (2024) studied this question.
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