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.