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