Harmonic Moments and Large Deviations for the Markov Branching Process with Immigration

Abstract

Let $\{X_t;t\ge 0\}$ be a Markov branching process with immigration, in which each particle has exponential lifetime distribution with parameter a and offspring law $\{p_k;k=0,1,2,\ldots \}$. The time interval of immigration follows an exponential distribution with parameter $\theta$. Let $p_0=0$ and $\lambda :=a(\sum _{k=1}^{\infty }kp_k-1)<\infty$. In this paper, we investigate the large deviation behavior of $\left\{ X_{t+s}/X_t;t\ge 0,s>0\right\}$ by studying the asymptotic behavior of harmonic moments $E[X_t^{-r}]$ for $r>0$. We obtain that there is a “phase transition" in rates depending on whether $\lambda r-a-\theta$ is less than, equal to or greater than 0.