Large deviations for a Poisson random indexed branching process

Consider a Galton–Watson process {Zn}{Zn} and an independent Poisson process {Nt}{Nt}, the continuous time process {ZNt}{ZNt} is a Poisson random indexed branching process. We show the large deviation results for P(ZNt≤ect)P(ZNt≤ect) and P(ZNt≥ect)P(ZNt≥ect).