Lower deviation probabilities for supercritical Markov branching processes with immigration

Let $\{Z(t);t\ge 0\}$ be a continuous-time supercritical branching process with immigration (MBPI) with the offspring mean $m(t)$. In this paper, we mainly research the lower deviation probabilities $\mathrm{P}(Z(t)=k_t)$ and $\mathrm{P}(0\le Z(t)\le k_t)$ with $k_t/e^{m(t)}\to 0$ as $t\to \mathrm{\infty }$. Moreover, we present the local limit theorem and some related estimates of the MBPIs. For our proofs, we use the well-known Cramér method to prove the large deviation of the sum of independent variables to satisfy our needs.