Large Deviation Rates for Supercritical Branching Processes with Immigration

Let $$\{X_n\}_0^{\infty }$$ { X n } 0 ∞ be a supercritical branching process with immigration with offspring distribution $$\{p_j\}_0^{\infty }$$ { p j } 0 ∞ and immigration distribution $$\{h_i\}_0^{\infty }.$$ { h i } 0 ∞ . Throughout this paper, we assume that $$p_0=0, p_j\ne 1$$ p 0 = 0 , p j ≠ 1 for any $$j\ge 1$$ j ≥ 1 , $$1 \varepsilon ), \ \ P\left( \left| \frac{X_{n+1}}{X_n}-m\right| >\varepsilon \Bigg |Y\ge \alpha \right) \end{aligned}$$ P ( Y n - Y > ε ) , P X n + 1 X n - m > ε | Y ≥ α for $$\varepsilon >0$$ ε > 0 and $$\alpha >0$$ α > 0 under various moment conditions on $$\{p_j\}_0^{\infty }$$ { p j } 0 ∞ and $$\{h_i\}_0^{\infty }.$$ { h i } 0 ∞ . It is shown that the rates are always supergeometric under a finite moment generating function hypothesis.