Precise Large Deviations for the Total Population of Heavy-Tailed Subcritical Branching Processes with Immigration

In this article, we focus on the partial sum $$S_{n}=X_{1}+\cdots +X_{n}$$ S n = X 1 + ⋯ + X n of the subcritical branching process with immigration $$\{X_{n}\}_{n\in \mathbb {N_{+}}}$$ { X n } n ∈ N + , under the condition that one of the offspring $$\xi $$ ξ or immigration $$\eta $$ η is regularly varying. The tail distribution of $$S_n$$ S n is heavily dependent on that of $$\xi $$ ξ and $$\eta $$ η , and a precise large deviation probability for $$S_{n}$$ S n is specified. (i) When the tail of offspring $$\xi $$ ξ is “lighter” than immigration $$\eta $$ η , then uniformly for $$x\ge x_{n}$$ x ≥ x n we have $$P(S_{n}-ES_{n}>x)\sim c_{1}nP(\eta >x)$$ P ( S n - E S n > x ) ∼ c 1 n P ( η > x ) with some constant $$c_{1}$$ c 1 and sequence $$\{x_{n}\}$$ { x n } , where $$c_{1}$$ c 1 is related only to the mean of offspring; (ii) when the tail of immigration $$\eta $$ η is not “heavier” than offspring $$\xi $$ ξ , then uniformly for $$x\ge x_{n}$$ x ≥ x n we have $$P(S_{n}-ES_{n}>x)\sim c_{2}nP(\xi >x)$$ P ( S n - E S n > x ) ∼ c 2 n P ( ξ > x ) with some constant $$c_{2}$$ c 2 and sequence $$\{x_{n}\}$$ { x n } , where $$c_{2}$$ c 2 is related to both the mean of offspring and the mean of immigration.