Large Deviation Rates for Supercritical Branching Processes with Immigration

Abstract

Let $$\{X_n\}_0^{\infty }$$ be a supercritical branching process with immigration with offspring distribution $$\{p_j\}_0^{\infty }$$ and immigration distribution $$\{h_i\}_0^{\infty }.$$ Throughout this paper, we assume that $$p_0=0, p_j\ne 1$$ for any $$j\ge 1$$ , $$1<m=\sum _{j=0}^{\infty } jp_j<\infty ,$$ and $$h_0<1$$ , $$0<a=\sum _{j=0}^{\infty } jh_j<\infty .$$ We first show that $$Y_n=m^{-n}(X_n-\frac{m^{n+1}-1}{m-1}a)$$ is a martingale and converges to a random variable Y. Secondly, we study the rates of convergence to 0 as $$n\rightarrow \infty $$ of $$\begin{aligned} P(\left| Y_n-Y\right|>\varepsilon ), \ \ P\left( \left| \frac{X_{n+1}}{X_n}-m\right| >\varepsilon \Bigg |Y\ge \alpha \right) \end{aligned}$$ for $$\varepsilon >0$$ and $$\alpha >0$$ under various moment conditions on $$\{p_j\}_0^{\infty }$$ and $$\{h_i\}_0^{\infty }.$$ It is shown that the rates are always supergeometric under a finite moment generating function hypothesis.