Large Deviations for Galton–Watson branching processes with immigration

Let $\{Z_n\}_{n\ge 0}$ be a Galton–Watson branching process with immigration in one particle. By definition, put
$$ Z_0 = 0,\quad Z_{n} = \sum_{j=1}^{1+Z_{n-1}}X_{n,j},\quad n\in\mathbb{N}. $$
Here random variables $X_{i,j}$ are independent identically distributed taking non-negative integer values. Put
$$ S_0 = 0,\quad S_n = \sum_{i=1}^{n} Z_i. $$
We obtain the exact asymptotics of large deviations probabilities for $S_n$ in the local form. In the subcritical case ($\mathbf{ E}X_{1,1}<1$) under small additional restrictions we obtain the local central limit theorem.