Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration

Abstract

Let $\left\{Y\left(t\right);t\ge 0\right\}$ be a supercritical continuous-time branching process with immigration; our focus is on the large deviation rates of $Y\left(t\right)$ and thus extending the results of the discrete-time Galton–Watson process to the continuous-time case. Firstly, we prove that $Z\left(t\right)=e^{-mt}\left[Y\left(t\right)-\left(\left(e^{m\left(t+1\right)}-1\right)/\left(e^m-1\right)\right)e^{a+m}\right]$ is a submartingale and converges to a random variable $Z$ . Then, we study the decay rates of $P\left(\left|Z\left(t\right)-Z\right|>\epsilon \right)\text{as\hspace{0.17em}}t⟶\infty$ and $P\left(\left|\left(Y\left(t+v\right)/Y\left(t\right)\right)-e^{mv}\right|>\epsilon |Z\ge \alpha \right)\text{as\hspace{0.17em}}t⟶\infty$ for $\alpha >0$ and $\epsilon >0$ under various moment conditions on $\left\{b_k;k\ge 0\right\}$ and $\left\{a_j;j\ge 0\right\}$ . We conclude that the rates are supergeometric under the assumption of finite moment generation functions.