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In this article we focus on the partial sum S₍=X₁++X₍ of the subcritical branching process with immigration \X₍\₍_+, under the condition that one of the offspring or immigration is regularly varying. The tail distribution of Sₙ is heavily dependent on that of and, and a precise large deviation probability for S₍ is specified. (i) When the tail of offspring is lighter than immigration, uniformly for x x₍, P (S₍-ES₍>x) c₁nP (>x) with some constant c₁ and sequence \x₍\, where c₁ is only related to the mean of offspring; (ii) When the tail of immigration is not heavier than offspring, uniformly for x x₍, P (S₍ ES₍>x) c₂nP (>x) with some constant c₂ and sequence \x₍\, where c₂ is related to both the mean of offspring and the mean of immigration.
Guo et al. (Tue,) studied this question.