Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration

In this article we focus on the partial sum Sₙ=X₁+⋯+Xₙ of the subcritical branching process with immigration ₙ_n∈N₊, under the condition that one of the offspring ξ or immigration η is regularly varying. The tail distribution of Sₙ is heavily dependent on that of ξ and η, and a precise large deviation probability for Sₙ is specified. (i)When the tail of offspring ξ is lighter than immigration η, uniformly for x≥ xₙ, P(Sₙ-ESₙ>x)~ c₁nP(η>x) with some constant c₁ and sequence ₙ\, where c₁ is only related to the mean of offspring; (ii) When the tail of immigration η is not heavier than offspring ξ, uniformly for x≥ xₙ,P(Sₙ ESₙ>x)~ c₂nP(ξ>x) with some constant c₂ and sequence ₙ\, where c₂ is related to both the mean of offspring and the mean of immigration.